The role of traction in membrane curvature generation

Abstract

Curvature of biological membranes can be generated by a variety of molecular mechanisms including protein scaffolding, compositional heterogeneity, and cytoskeletal forces. These mechanisms have the net effect of generating tractions (force per unit length) on the bilayer that are translated into distinct shapes of the membrane. Here, we demonstrate how the local shape of the membrane can be used to infer the traction acting locally on the membrane. We show that buds and tubes, two common membrane deformations studied in trafficking processes, have different traction distributions along the membrane and that these tractions are specific to the molecular mechanism used to generate these shapes. Furthermore, we show that the magnitude of an axial force applied to the membrane as well as that of an effective line tension can be calculated from these tractions. Finally, we consider the sensitivity of these quantities with respect to uncertainties in material properties and follow with a discussion on sources of uncertainty in membrane shape.

FIGURE 1:

Curvature generation in biological membranes (adapted from Chabanon et al., 2017). Membrane curvature is controlled by different physical inputs including (A) protein-induced spontaneous curvature and (B) forces exerted by the cytoskeleton.

FIGURE 2:

Schematic representing the axisymmetric coordinate system used for calculating curvature and traction. ω is the membrane surface area bounded by a curve ∂ω, f is an externally applied force per unit area on the membrane, n is the normal vector to the surface, ν is the tangent to the surface in the direction of increasing arc length, e_r and e_z are unit vectors in radial and axial directions, τ is a unit vector tangent to the boundary in the direction of the surface of revolution, ψ is an angle made by the tangent with respect to the horizontal, θ is the angle of revolution, s is the arc-length parameterization, p is the transmembrane pressure difference, and are radial and axial tractions along the curve of revolution, respectively. Inset shows that pressure opposes traction and external force in both the radial and axial directions.

FIGURE 3:

Analysis of normal and tangential traction for membrane tethers. (A) Normal traction distribution along four membrane tether shapes obtained by applying a point load of the specified magnitude at the pole, λ₀ = 0.02 pN/nm, κ = 320 pN·nm. (B) Magnitude of axial force as a function of tether height, showing an exact match between the force (Eq. 5) calculated from the traction distribution and obtained directly from the simulation. (C) Tangential traction distribution along the membrane shapes shown in A. (D) Energy per unit length calculated using Eq. 6 along the four membrane shapes shown in A. The dashed lines outline the equilibrium geometry for a membrane cylinder .

FIGURE 4:

Comparison of normal and tangential tractions between multiple mechanisms of membrane tether formation. (A) EM image of an endocytic PM invagination in a bzz1∆rvs167∆ yeast cell (Kishimoto et al., 2011). Top, original EM image; bottom, EM image with traced membrane shape (white). (B) Simulated membrane shape obtained by application of a point force (brown), λ₀ = 0.02 pN/nm, κ = 320 pN·nm. (C) Normal traction distribution along the membrane shape in B. (D) Tangential traction distribution along the membrane shape in B. (E) EM image of an endocytic PM invagination in a wild-type yeast cell (Kishimoto et al., 2011). Top, original EM image; bottom, EM image with traced membrane shape (white). (F) Simulated membrane shape obtained by application of an anisotropic spontaneous curvature (green) along the tubular section of a membrane tether, λ₀ = 0.02 pN/nm, κ = 320 pN·nm, C = −0.01 nm⁻¹, D = 0.01 nm⁻¹. (G) Normal traction distribution along the membrane shape in F. (H) Tangential traction distribution along the membrane shape in F. (I) Electron tomography image of an endocytic invagination in budding yeast (Kukulski et al., 2012). Top, original EM image; bottom, EM image with traced membrane shape (white). (J) Simulated membrane shape obtained by application of a point force (brown) against an equivalent pressure to the membrane tension in B, λ₀ = 0.02 pN/nm, κ = 320 pN·nm, p = 0.3 kPa. (K) Normal traction distribution along the membrane shape in J. (L) Tangential traction distribution along the membrane shape in J.

FIGURE 5:

Analysis of budding due to protein-induced spontaneous curvature and calculation of line tension. Simulations were conducted with (A = 10,053 nm²) spontaneous curvature at the center of an initially flat patch increasing from C = 0 to C = 0.032 nm⁻¹, λ₀ = 0.02 pN/nm, κ = 320 pN·nm, p = 0 (Hassinger et al., 2017). (A) Membrane shapes for three different spontaneous curvature distributions, with the value of C indicated in the red region and zero in the black region. (B) Normal traction along the membrane for the shapes shown in A. (C) Tangential traction distribution along the shapes shown in A. (D) Energy per unit length distribution for the three different shapes. The dashed line circles outline spheres with mean curvatures H = 0.032 nm⁻¹ (smaller circle) and H = 0.025 nm⁻¹ (larger circle).

FIGURE 6:

Change in energy per unit length and its components at the interface with increasing spontaneous curvature. Two regimes are observed: a surface tension–dominated regime for small value of spontaneous curvature and a curvature gradient–dominated regime for large values of spontaneous curvature. The membrane configurations are shown for two spontaneous curvatures: C = −0.02 nm⁻¹, where energy per unit length at interface is zero; and C = −0.025 nm⁻¹, where energy per unit length is maximum. The red domains show the region of spontaneous curvature for the corresponding shapes.

FIGURE 7:

Comparison of normal and tangential tractions between two different mechanisms of membrane budding. (A) EM image of COPII budding from the ER in green algae (Robinson et al., 2015). Left, original EM image; right, EM image with traced membrane shape. Red, COPII coat; white, bare membrane. (B) Simulation of bud formation on a hemispherical cap using a constant spontaneous curvature (C = −0.046 nm⁻¹, red). (C) Normal traction distribution along the membrane shape in B. A large negative normal traction can be seen at the neck of the formed vesicle. (D) Tangential traction distribution along the membrane shape in B. There is a change in the sign of the tangential traction before and after the bud neck. (E) Bright-field microscopy image of a budding yeast (Mozdy et al., 2000). Left, original EM image; right, EM image with traced membrane shape. Brown, contractile ring at the bud neck. (F) Simulation of bud formation on a hemispherical cap with a constant radial force (F_r = 6.2 pN, yellow) that locally constricts the hemisphere to form a bud. (G) Normal traction distribution along the membrane shape in F. There is a positive normal traction at the vesicle neck in response to the applied force. (H) Tangential traction distribution along the membrane shape in F.

FIGURE 8:

Parametric sensitivity analysis to material properties. Axial force (Eq. 5) and energy per unit length (Eq. 6) were calculated for a variation in the bending rigidity κ and membrane tension λ₀ both in membrane tubes (A, B) and buds (C, D). Dashed lines indicate 10% error. λ_mean = 0.02 pN/nm, κ_mean = 320 pN·nm, −(F_z)_mean = 18 pN (corresponding to a tube of height 300 nm in Figure 3), ξ_mean= 6.13 pN (corresponding to a spontaneous curvature of 0.0276 nm⁻¹ in Figure 5). The sensitivity analysis was performed in two ways. 1) Sensitivity to shape and material property by running multiple simulations corresponding to the different parameter values (A, C), followed by an error calculation with respect to the mean value. 2) Sensitivity to only material property by using a range of parameter values during calculation of axial force (Eq. 5) and energy per unit length (Eq. 6) for a single simulation (mean) (B, D). (A) Sensitivity to shape and material property in a membrane tube. (B) Sensitivity to only material property in a membrane tube. (C) Sensitivity to shape and material property in a membrane bud. (D) Sensitivity to only material property in a membrane bud.