Schematic illustrations of the constant area model (a) and the constant curvature model (b) for CME. Blue: plasma membrane, yellow: clathrin coat, black dashed line: curvature of the clathrin coat

Evolution of membrane morphology and phase diagram of vesiculation in the coating area vs. intrinsic curvature (a0, c0) parameter space. (a) Membrane shapes at different stages of invagination and definition of some variables used in this paper. We define the distance from the axisymmetric axis to the edge of the coating area as Rcoat, the radius of the tangential curvature circle at the tip of the shape as Rt, and the distance from axisymmetric axis to the boundary as Rb. The maximum tangential angle of the cross section contour is Ψmax. (b,c) Tip radius Rt vs. maximal angle Ψmax for the constant area model in (b) and for the constant curvature model in (c). Dotted lines in (b) denote the analytical solutions. Insets show the ratio of the tip radii Rt at Ψmax = 150° and Ψmax = 30°. The inset dark dots denotes the numerical results and the red line is the analytical solution (See Appendix 5). (d,e) Coat radius Rcoat vs. maximal angle Ψmax for the constant area model in (d) and for the constant curvature model in (e). Dotted lines in (d) denote the analytical solutions. Insets show the ratio of the coat radius Rcoat at Ψmax = 150° and Ψmax = 90°. The inset dark dots denotes the numerical results and the red lines are the analytical ones (See Appendix 5). (f) Vesiculation diagram in the phase space of (a0, c0). Each horizontal line represents a path of the constant curvature model and each vertical line represents a path of the constant area model. Each path terminates when Ψmax = 150°. The solid grey lines represent contours of Ψmax = 30°, 60°, 90°, 120°, 150°, respectively. The solid black line is the analytical results for the vesiculation line (See Appendix 4). The dashed black line is a random-picked straight line connecting the origin and the vesiculation boundary. The intersections of the dashed black line and the gray lines are plotted in orange dots and they are the coordinates of (a0, c0) where the five shapes in (a) are located. Shapes are arranged in an increasing order of Ψmax in (a). (b-e) Parameters with a bar over them (left vertical axes) are normalized to be dimensionless, and the dimensional parameters (right vertical axes) are calculated by one of the typical fitting values L0 = 40nm (Figure 5)

Free energy evolution in the constant curvature and constant area models when accounting for one of either the polymerization energy term Ea = −μa0 or the curvature generation energy term Ec = −va0c0 (type 1) and . (a,b) Free energy landscape of the modified constant curvature model where with polymerization energy Ea = −μa0 in (a) and the corresponding phase diagram in the phase space of (c0, μ) in (b). The rightmost endpoint of each curve is the vesiculation point where Ψmax = 150°. The red line in (a) and red dots in (b) correspond to pathways with minimum free energy Etot appearing at a point other than the vesiculation point on the Etot − ā0 curve. The orange line in (a) and the orange dots in (b) correpsond to pathways where the vesiculation point is the minimum free energy point, but an energy barrier still exists. The green line in (a) and green dots in (b) correspond to vesiculation pathways without an energy barrier. The energy barrier ΔEtot is defined as the energy difference between the maximum point and the first minimum point before the maximum. ΔEtot of the orange curve is shown in (a) as a typical example. The gray dots in the left-hand side of (b) correspond to pathways that numerically fail to reach the vesiculation point when ā0 reaches its upper limit 10. (c,d) Free energy landscape of the modified constant area model where ā0 = 2 with curvature generation energy Ec = −va0c0 in (c) as a function of the intrinsic curvature c0 and the corresponding phase diagram in the phase space of (a0, v) in (d). The gray dots in the left-hand side of (d) correspond to pathways that numerically fail to reach the vesiculation point when reaches its upper limit 5. (e,f) Free energy landscape of the modified constant area model where ā0 = 2 with in (e) and the corresponding phase diagram in the phase space of (a0, v) in (f). The gray dots in the left-hand side of (f) correspond to pathways that numerically fail to reach the vesiculation point when reaches its upper limit 5. (a-f) The parameters with a bar over them are normalized to be dimensionless, and the dimensional parameters are calculated using one of the typical fitting values L0 = 40nm (Figure 5). (b,d,f) The black line separates the region wiht gray dots (which did not numerically reach vesicultion) from the other regions. The dotted, dash-dotted and solid gray lines respectively represent the paths where ΔEtot = 1kBT, ΔEtot = 10kBT, ΔEtot = 100kBT.

Vesiculation phase diagram when accounting for both the polymerization energy μ and the reorganization energy v. (a,b) Free energy landscape for Model(1,2) (i.e. with reorganization energy in (a) and Model(1,1) (i.e. with reorganization energy in (b). Thick black lines are the analytical solutions for the vesiculation boundary. Thin rainbow-colored lines visually represent the energy landscape (values of the color bar are for the dimensionless free energy scale). Thick colored-lines represent pathways that stream along the negative gradient of the free energy landscape in the phase space starting from the origin (i.e. no clathrin assembled and no curvatuve). Our model shows that only a subset of suitable (μ, v) values create pathways that lead to vesiculation, i.e. that reach the thick black line (thick green lines in the third and fourth panels). The orange and red curves are pathways that fail to reach vesiculation. The red curve does not produce any curvature, while the orange curve generates a small curvature but never leads to vesiculation. (c,d) Phase diagrams for Model(1,2) in (c) and Model(1,1) in (d) show the relationship between pathway types and the (μ, v) values. The colors of the dots correspond to the same types of pathways as represented by thick colored lines in (a,b). Our results show that larger μ or v values lead to an easier vesiculation. Parameters with a bar over them are normalized to be dimensionless, and the dimensional parameters are calculated using one of the typical fitting values L0 = 40nm (Figure 5).

Comparison between our theory and experimental data from mammalian cells. (a) Parameter fit of the best of the four models (constant curvature model, constant area model, Model(1,1), Model(1,2)) to obtain the minimum error ϵ. Fitting procedure of Model(1,1) and Model(1,2) consider the total free energy Etot = Eb + Et + Ea + Ec, while the fitting of the constant area model and the constant curvature model consider Etot = Eb + Et. The optimized parameters are ā0 ∈ [0, 10] for the constant area model, for the constant curvature model, and for model(m,n), and L0 ∈ [10nm, 100nm] within an interval of 10nm in the four models. We only assign fitting errora to the parameter sets that lead to vesiculation and only plot the error figure for the best L0. (b) Vesiculation pathways with minimum fitting error in the four models. In each model, we use the best L0 value from (a) to obtain the dimensional scale of the (a0, c0) phase space. (c) Comparison of model fits and experimental data for three geometric features: neck width, tip radius (Rt) and invagination depth. Neck width is calculated as the distance between the left and right parts of the shape for Ψmax = 90°, and the invagination depth is measured as the height from the base to the tip of the invagination. (d) Comparison between the model-predicted shapes and the experimental shapes. Experimental membrane shapes for mammalian cells are grouped according to their maximum angle as a proxy for the different stages of CME. The number of experimental shapes falling in a certain Ψmax range is defined as n. The black lines are the average experimental shapes after symmetrization. The model-predicted shapes are calculated by the midpoint value of each Ψmax interval. (c,d) The curves predicted by theory are shown with colored lines, and experimental data is shown with gray dots and black lines. Parameters with a bar over them are non-dimensionalized. The detailed procedure to treat the experimental data can be found in Appendix 3.

Tip radius of vesiculaion shapes (Rves) in Model(1,2). The colored region shows the (μ, v) sets that lead to vesiculation, and brighter colors correspond to larger Rves. Decreasing assembly strength μ or increasing reorganization strength v might lead to vesiculation of different vesicle sizes. An example from (μ, v) = (13.3 × 10−3kBT · nm−2, 8.8kBT) to the vesiculation region is marked by arrows, red dots and corresponding vesicle shapes. The characteristic length L0 = 30nm is used in the calculation.

Vesiculation pathways and free energies in the different models studied in this paper. (a,b) Ψmax for the constant area model and the constant curvature model. (c,d) Free energy for the constant area model and the constant curvature model. (a-d) Colored lines represent vesiculation pathways for a fixed a0 or c0 specified in the legend. Solid lines represent states that have the minimum free energy, while dotted lines represent energetically possible states but are metastable. For certain range of a0 or c0, a single of a0 or c0 corresponds to multiple values of Ψmax. In the free energy diagram, this is reflected in the Gibbs triangle, which means there will be a snap-through transition of Ψmax. (e) Vesiculation boundary in the (a0, c0) phase diagram. Each green line represents a pathway in one of the two models (horizontal lines for the constant curvature model, vertical lines for the constant area model). Each line stops when vesiculation occurs (i.e. Ψmax = 150°) according to the numerical simulations of the model. The black line represents the vesiculation boundary as determined analytically, i.e. when (See Appendix 4). The solid orange line represents (a0, c0) values for which a Ψmax gap exists. The region between the two dotted lines represent that a single pair of (a0, c0) corresponds to three values of Ψmax, and the solid line represents the critical line at which a snap-through transition of the shape will occur. Note that the vesiculation boundaries detemined numerically or analyticaly poorly match to each other because the ratio kbare/kcoat is small and Equation 30 is not satisfied (See Appendix 4).

Influence of Rb on the shape parameters in the constant area and constant curvature models. (a and d) Neck width. (b and e) Tip radius (note that the neck width is ill-defined when Ψmax is small, so we restrict the abscissa range, and the membrane height is z(u) at u = 1). (c and f) Membrane height. (a-c) Constant area model. (d-f) Constant curvature model. Default parameters used in this figure:.

List of default parameters in the model.

symmetrization of the experimental profile. (a) The experimental profile curve is divided into a left part and a right part he profile. Each part is interpolated onto the same rescaled mesh points ui = si/Si, where si is the arclength t, Si is the total arclength to the last point and i = 1, 2 indicates the left or right section, repectively. The length of d the translucent red line is S2. (b) Symmetrized experimental profile by taking the average of the left part and ed arclength.

Model parameters and their units.

Approximate spherical cap model at small Ψmax (left) and large Ψmax (right). The radius of the spherical cap is R, the max tangential angle of the sphere is Ψmax, equaling to the angle at the base. Other definitions are the same as in the main text. R = Rt > Rcoat, θ = Ψmax when Ψmax < 90° and R = Rt = Rcoat, θ = Ψmax when Ψmax ≥ 90°.