We model the membrane patch of the CCP (clathrin-coated pit) as a surface which is rotationally symmetric with respect to the z-axis. The shape of the membrane is parameterized with the meridional curve {r(s), z(s)}, where s denotes the arc length along the curve. The bending energy of the membrane (together with the clathrin coat) assumes the Helfrich model (Helfrich, 1973) where κ denotes the bending rigidity of the CCP, C₁ and C₂ denote the principal curvatures of the surface, C₀ denotes the intrinsic curvature of the membrane induced by the clathrin coat. To model a finite area of the clathrin coat, we assume the intrinsic curvature C₀ spatially varies as where a denotes positions on the membrane. Here, we choose a to be the surface area calculated from the tip of the membrane, and C₀ equals c₀ for area a < a₀, and rapidly drops to zero when a > a₀. The parameter α controls the sharpness of the drop. In the constant area model, we vary the intrinsic curvature c₀ but keep the coating area a₀ constant, while in the constant curvature model, we vary the coating area a₀ but keep the intrinsic curvature c₀ constant. As a result of the clathrin coat, the bending rigidity κ also varies as where κ_coat and κ_bare denote the bending rigidity of the clathrin-coated membrane and bare membrane, respectively. Besides the bending energy, the membrane tension contributes to the free energy in the form of where σ_e denotes the membrane tension at the base and A denotes the surface area of the membrane patch within a fixed radius of R_b. The membrane tension σ_e and bending rigidity κ define a characteristic length (Derényi et al., 2002). The total free energy E_tot = E_b + E_t is a functional of the membrane shape. We emphasize that the constant area model refers to the assumption that the clathrin-coated area a₀ remains fixed. Meanwhile, the membrane tension σ_e at the base is held constant, allowing the total membrane area A to vary in response to deformations induced by the clathrin coat. Given a coating area a₀ and an intrinsic curvature c₀, we numerically solve the variational equations of the energy functional to obtain membrane shapes that minimizes E_tot. This approach is based on the assumption that the membrane remains in mechanical equilibrium throughout the process of clathrin assembly, which is justified by the disparity between the timescales of membrane relaxation and clathrin assembly. The characteristic relaxation time of a lipid membrane is given by , where µ ≈ 5 × 10⁻⁹ N ⋅ s ⋅ m⁻¹ is the membrane viscosity Arroyo and DeSimone (2009), R₀ ≈ 50 nm is the vesicle size Avinoam et al. (2015), and κ ≈ 20 k_BT is the bending rigidity Yuan et al. (2021). Substituting these values yields a relaxation time of τ ≈ 1.5 × 10⁻⁴ s, which is several orders of magnitude shorter than the clathrin assembly timescale which is approximately one minute. Therefore, it is reasonable to assume that the membrane rapidly equilibrates and remains in mechanical equilibrium during the assembly process. More detailed descriptions of the model can be found in Appendix 1.

We rescale lengths by the characteristic length L₀ and energies by 2πκ_bare. Dimensionless quantities are denoted with a bar. The dimensionless tension energy, defined as Ē_t = E_t/(2πκ), simplifies to Ē_t = Ā/2, where Ā = A/(2πL²) is the dimensionless membrane area. Therefore, the dimensionless membrane tension becomes a constant 1/2. When presenting data, dimensionless quantities are shown on the left axes, and the corresponding dimensional values are shown on the right axes.

Vesiculation requires assembly of a clathrin coat on the membrane, as well as curvature generation from the clathrin coat. A vesiculation process defines a pathway in the phase space (a₀, c₀) of the clathrin coat area a₀ and the intrinsic curvature c₀ of the coat. The constant area model and the constant curvature model are pathways that are made of a vertical line and a horizontal line. Besides these two extreme cases, there is a continuum spectrum of pathways with simultaneously increasing coating area a₀ and intrinsic curvature c₀ that could lead to vesiculation. Along the pathway, the membrane evolves from a flat shape to a dimple shape, and finally to an Ω-shape, as shown in Figure 2. Hereafter we use the maximal tangential angle ψ_max of the membrane as an indicator of the progression of vesiculation - when the membrane is flat, ψ_max = 0^°, and when the membrane becomes spherical, ψ_max = 180^°. In our simulation, the neck becomes extremely narrow before ψ_max = 180^°. Therefore, we consider vesiculation occurs when ψ_max reaches 150^° (Figure 2a).

First, we analyze the difference between the two models in terms of membrane morphology evolution along their pathways. We fit a circle around the membrane tip and use the radius R_t of the circle to characterize the curvature of the membrane at the tip. When R_t is plotted against the maximal angle ψ_max, both models show that R_t decreases with increasing ψ_max (Figure 2b and c). We stress that even though the intrinsic curvature c₀ is fixed in the constant curvature model, it does not imply the tip radius along the vesiculation pathway is a constant. When the coating area is small, the geometric curvature at the membrane tip remains small and differs from the intrinsic curvature. Tip radius in the constant curvature model decays more steeply with ψ_max than in the constant area model. This difference becomes obvious when one plots the ratio of the tip radius at ψ_max = 150^° and ψ_max = 30^° (Figure 2b and c insets). For the constant area model, the ratio approximately equals to a constant 0.268 regardless of the coating area a₀, which agrees well with the analytical result (See Appendix 5). For the constant-curvature model, the ratio remains close to 1 only at small values of c₀, as expected from the schematic representation of the model in Figure 1. However, as c₀ increases, the deviation from this idealized picture becomes increasingly pronounced.

Another difference between the two models is the evolution of the projected area of the clathrin coat on the substrate. We use the maximal radius R_coat of the membrane within the clathrin-coated area as the indicator of the projected area (Figure 2a). In the constant area model, R_coat decreases with increasing ψ_max, while in the constant curvature model, R_coat increases with ψ_max and reaches a plateau (Figure 2d and e). The ratio R_coat (150^°)/R_coat (90°) is about 0.732 in the constant area model and around 1 in the constant curvature model (Figure 2d and e, insets). The analytical curves of R_t and R_coat against ψ_max also fit perfectly with numerical solutions in the constant area model (Figure 2b and d, compare dotted and solid curves). Our calculations therefore demonstrate that the two models exhibit clear differences in the evolution of R_t and R_coat at the beginning of endocytosis (i.e. when ψ_max is small) which can be determined from shapes of endocytic pits obtained experimentally.

To demonstrate how the coating area a₀ and the intrinsic curvature c₀ of the clathrin coat influence the membrane morphology, for each pair of (a₀, c₀), we calculate the corresponding membrane shapes and plot the contour lines for ψ_max which indicate the stage of endocytosis (Figure 2f). The contour line with ψ_max = 150^° represents the critical line where vesiculation occurs. The line can be well fitted by the analytical expression (Figure 2f, thick black line, see Appendix 4). It implies that a small intrinsic curvature of the clathrin coat is able to induce vesiculation of the membrane with a large clathrin coat. We find that the distances between contour lines for higher values of ψ_max is smaller than those for lower values of ψ_max. It means that at the late stage of endocytosis, a small change in a₀ and c₀ could result in a more dramatic change in the membrane shape than that at the early stage. This trend is demonstrated clearly from the orange dots in Figure 2f, which correspond to shapes in Figure 2a.

Note that in order to produce the phase diagram (Figure 2f), it requires that the bending rigidity of the clathrin coated area κ_coat is significantly larger than κ_bare in the uncoated area. If κ_coat is comparable with κ_bare, there exists a region in the phase diagram in which a single (a₀, c₀) corresponds to three possible membrane shapes (See Appendix 2—Figure 1a-d). Physically, it implies a discontinuous transition in the membrane shape along a path that passes through this region, and a gap in the maximal angle ψ_max would appear. Because in experiments, a wide spectrum of ψ_max are observed and no gap in ψ_max is found (Avinoam et al., 2015), we keep κ_coat much greater than κ_bare for the rest of the paper. In this regime, the membrane shapes evolve continuously along any pathway that connects the origin (0, 0) with a point on the critical vesiculation curve.

In the previous section, we take curvature generation in the constant area model and clathrin assembly in the constant curvature model for granted, so that the coating area a₀ and the intrinsic curvature c₀ are imposed, such that the physical forces behind clathrin assembly and curvature generation are ignored. However, the bending energy E_b and the tension energy E_t typically increase with a₀ and c₀ along a vesiculation pathway. Therefore, vesiculation will be energetically unfavorable if no additional free energy sources are provided. In this section, we extend the model to include free energy terms for the assembly of the clathrin coat and its reorganization for curvature generation.

To describe the assembly of the clathrin coat in the constant curvature model, we introduce where µ denotes the effective surface binding energy density of clathrin molecules with the membrane. This term reduces the free energy with increasing a₀, therefore, driving the assembly of clathrin. We can identify three types of free energy curves for different assembly strength µ: (1) When µ is small, the total free energy E_tot = E_b + E_t + E_a as a function of the coating area a₀ has two local minima, with the lowest one at a small a₀ and the other one at the maximum a₀ where vesiculation occurs (Figure 3a, red curve). The minima are separated by an energy barrier that is significantly higher than the thermal energy k_BT and the clathrin coat would assemble to a small area and halt. (2) With increasing µ, the lowest free energy minimum is shifted to the vesiculation point, but the energy barrier still exists and the clathrin coat remains small (Figure 3a, orange curve). (3) Vesiculation could happen for large enough µ such that the energy barrier vanishes and the free energy E_tot monotonically decreases with a₀ (Figure 3a, green). Based on the above analysis of the energy landscape we construct the phase diagram of the constant curvature model with clathrin assembly in the phase space of (c₀, µ) and classify the points into four types. Besides the three types mentioned above, when the intrinsic curvature c₀ is small, increasing a₀ to its maximum value (10₅nm₂) cannot produce vesiculation. (Figure 3b, gray region). The critical assembly energy density µ at which the energy barrier vanishes is found to increase with the intrinsic curvature c₀ (Figure 3b, interface between the green region and the orange region), which implies that a larger assembly strength of clathrin coat µ is needed to complete vesiculation if the clathrin coat has a higher intrinsic curvature c₀. When comparing the contour lines of the energy barrier ΔE_tot = 1k_BT and ΔE_tot = 10k_BT (Figure 3b, dotted curve and dash-dotted curve), the gap between them increases with c₀, which means that the energy efficiency is reduced with c₀ in the sense that, for larger c₀, a larger increase in µ is needed to reduce the same amount of free energy.

We next consider curvature generation in the constant area model. As the molecular mechanisms of curvature generation of the clathrin coat remains debated, we introduce a phenomenological model in which the free energy has the general form, where ν denotes the strength of curvature generation, m and n are two positive numbers that are associated with the molecular mechanisms of curvature generation. The free energy E_c in Equation 6 decreases with increasing c₀, therefore, driving curvature generation. We set m = 1 such that E_c is proportional to the coating area. Note that m cannot be zero, otherwise, E_c only depends on the intrinsic curvature c₀ and can be nonzero even when the coating area is zero. As for the power n of the intrinsic curvature c₀, we set n = 1 or 2 (called Model(1,1) and Model(1,2), respectively). Physically, Model(1,2) implies cooperativity in the curvature generation such that the reduction of free energy per increase of unit curvature is proportional to the current curvature, i.e., ΔE_c ∝ −c₀Δc₀, while in Model(1,1), the reduction of free energy per increase of unit curvature is independent of current curvature. For Model(1,1), when ν is small, the total free energy E_tot = E_b + E_t + E_c as a function of the intrinsic curvature c₀ has two minima, the lowest one at a small positive c₀ and the other one at the maximum c₀ where vesiculation occurs (Figure 3c, red curve). Further curvature generation is strictly limited by the high energy barrier (sometimes more than 100k_BT) between the two minima. With increasing ν, the lowest minimum shifts to the vesiculation point, but the energy barrier still prevents curvature generation (Figure 3c, orange line). For a large enough ν, the free energy monotonically decreases with c₀ and the curvature generation proceeds until vesiculation occurs (Figure 3c, green curve). When the coating area a₀ is very small, vesiculation fails to occur even when the intrinsic curvature is increased to its maximum value (0.125nm⁻¹) (Figure 3c, gray region). In the phase space of (a₀, ν), the critical value of ν where the energy barrier vanishes increases with the coating area a₀ (Figure 3d, interface between the orange region and the green region), which implies that a larger clathrin coat needs a stronger strength of curvature generation to complete vesiculation.

Model(1,2) has similar free energy landscape as Model(1,1) (Compare Figure 3c and e, d and f). However, in Model(1,2), for very small ν, the lowest free energy minimum is strictly pinned at c₀ = 0, which implies no spontaneous curvature generation. In contrast, the minimum is at a small positive c₀ in Model(1,1), which indicates slight curvature generation.

In this section, we combine the assembly energy Equation 5 and the curvature generation energy Equation 6 together and calculate the total free energy E_tot (a₀, c₀) = E_b +E_t +E_a +E_c as a function of both the coating area a₀ and the intrinsic curvature c₀. A pathway from the origin can be constructed by the descent along the negative gradient of the free energy landscape −∇E_tot. In Figure 4a we show the free energy landscape for a fixed assembly strength and varied reorganization strength for model(1,2). When ν is small, the energy contour lines near the origin are kinked, which represents an energy barrier that prevents the path from going up, i.e. from generating curvature. The path extends horizontally and terminates on the a₀ − axis (Figure 4a, first column, red curve). With increasing ν, the kinked contour lines shift towards larger a₀ and the path can be lifted up to in the middle and drops to the a₀ − axis in the end (Figure 4a, second column, orange curve). Beyond a critical ν, the energy barrier vanishes and the path bends up and terminates on the vesiculation curve (Figure 4a, third column, green curve). Further increasing µ leads to the path lifting up at a smaller coating area a₀ (Figure 4a, fourth column, green curve). The path goes horizontally first and is later lifted up, which resembles the path of the constant area model. The three types of pathways are classified into three colored regions in the phase diagram of (µ, ν), which represent complete vesiculation (green), partial vesiculation (orange) and no vesiculation (red), respectively (Figure 4c).

The free energy landscapes of Model(1,1) dramatically differs from Model(1,2). When ν is small, the energy gradient is strongly biased towards the horizontal direction, and the path extends horizontally with little or no curvature generation (Figure 4b, first column). For an intermediate ν, the path first goes towards the top right direction until and then slowly bends down and extends towards large coating area along a valley formed in the energy landscape, which corresponds to membrane shapes with a small dimple (Figure 4b, second column). For large enough ν, the path shoots nearly straightly towards the top right direction before it reaches the vesiculation line (Figure 4b, third column). Further increasing ν makes the path more straight and terminates at a smaller coating area (Figure 4b, fourth column). The (µ, ν) phase diagram shows the parameter regions that lead to complete, partial or no vesiculation for Model(1,1) (Figure 4d).

The constant area model and the constant curvature model represent two extreme pathways of membrane vesiculation. We have found constant-area-like pathways in Model(1,2) and straight-line-like pathways in Model(1,1). In order to understand which model is the most plausible, we compare membrane shapes predicted by the models with the membrane profiles obtained by electron microscopy in (Avinoam et al., 2015). The fitting error ϵ of a vesiculation path reflects the relative difference between the model-predicted geometric features along the path and the rolling median of the corresponding experimental data (Figure 5 and Appendix 3). The fitting geometric features include neck width, tip radius, and invagination depth (Figure 5c). We draw the corresponding optimum energy paths that minimize the fitting error (Figure 5b), and compare the best model-predicted shapes with the experimental ones (Figure 5d).

For Model(1,2) and Model(1,1), the fitting parameters include the polymerization strength and the reorganization strength which together determine the vesiculation pathway, as well as the characteristic length L₀ which scales the size of the membrane. For Model(1,2), the best fits are obtained for and L₀ = 30nm (Figure 5a). The resulting path moves horizontally at first and then bents up vertically (Figure 5b), which resembles the behavior of the constant area model. For Model(1,1), the best fitting parameters are and L₀ = 40nm (Figure 5a). The fitting path is almost a straight line towards the vesiculation line (Figure 5b). The optimum fitting error of Model(1,2) (ϵ = 0.14) is slightly better than that of Model(1,1) (ϵ = 0.17).

We also perform the fitting procedure to the constant area model and find the optimum parameters are a₀ = 1.69 × 10⁴nm² and L₀ = 30nm. For the constant curvature model, the best fitting parameters are c₀ = 0.043nm⁻¹ and L₀ = 50nm. The minimum fitting error of the constant curvature model (ϵ = 0.28) is exactly twice as large as that of the constant area model (ϵ = 0.14) (Figure 5a and b). So considering the fitting error and the pathway in (a₀, c₀) phase diagram, we raise the conclusion that the experimental vesiculation process probably favors constant-area-like pathways.

When comparing the model-predicted geometric features with the rolling median of the experimental data, we find that the four models fit almost equally well the experimental data for the neck width(Figure 5c left). Model(1,2) and the constant area model predict very similar results such that the curves almost overlap with each other (Figure 5c, red curve and orange curve). The predictions of these two models match the rolling median of the experimental data best. The constant curvature model strongly deviates from the rolling median of the experimental tip radius, particularly in the early stage of vesiculation when ψ_max < 90^°.

To facilitate comparison between the axisymmetric membrane shapes predicted by the model and the non-axisymmetric profiles obtained from electron microscopy, we apply a symmetrization procedure to the experimental data, which consist of one-dimensional membrane profiles extracted from cross-sectional views, as detailed in Appendix 3 (see also Fig. 1). Then, we average the symmetrized profile within an interval of ψ_max ∈ [ψ₀ −10^°, ψ₀ +10^°] and overlay the averaged profile with model-predicted shapes for ψ_max = ψ₀ (Figure 5d). At the early stage when the membrane exhibits a dimple shape (0^° < ψ_max < 60^°), the membrane morphology predicted by the constant curvature model is distinguishable from the other three models, particularly when looking at the tip radius. At the late stage when the membrane exhibits an Ω-shape, i.e., ψ_max > 90^°, the difference in shape between models is mainly manifested in the invagination depth. The constant curvature model and Model(1,1) mainly predict a deeper invagination depth than the symmetrized experimental profile, while the constant area model and Model(1,2) usually give much better fitting.

In this paper, we have constructed a physical model to describe how curvature generation and clathrin assembly are interrelated during the vesiculation process in CME. Previous experiments have reported three groups of clathrin coated pits, which are plaques, abortive pits, and pits that lead to vesiculation, according to their structural and dynamic properties (Maupin and Pollard, 1983; Ehrlich et al., 2004; Loerke et al., 2009; Saffarian and Kirchhausen, 2009; Kirchhausen, 2009; Lampe et al., 2016).

In Figure 4, we show that depending on the clathrin assembly strength µ and its reorganization strength ν, the pathway might end up with three possible final shapes: (i) a flat membrane with no curvature generation, (ii) a nearly flat membrane with small curvature generation, (iii) a spherical cap that leads to vesiculation. They essentially correspond to the three types of clathrin-coated pits found in experiments. Based on the phase diagram of Model(1,2) (Figure 4c), the difference between the three groups comes from the difference in the assembly and reorganization strengths of clathrin molecules. Furthermore, Model(1, 2) predicts that at the boundary between the type (iii) region and the type (ii) region, the reorganization strength ν increases with the assembly strength. This result has important implications to explain why large plaques are commonly observed in experiments. The large area of the plaques are due to the strong assembly strength µ. However, for these plaques to go to vesiculation, a strong reorganization energy ν is also needed. Therefore, the combination of a strong µ and weak ν leads to the formation of large plaques. Model(1,2) predicts that a plaque or an abortive pit can be transformed into a vesicle by either increasing the reorganization strength or reducing the assembly strength (Figure 6). The former ends up with a large vesicle and the latter ends up with a small vesicle. This can be used as a test of our theory with experiments to modify the binding affinity of clathrin molecules with adaptor proteins on the membrane. Weakening the affinity might increase the portion of vesicles and reduce the portion of plaques, though the vesicles would become smaller.

In Figure 5, we show the best fitting results for all the four models and find that Model(1,2) produces better fits than Model(1,1), which suggests the existence of cooperativity in the curvature generation process. In particular, if curvature generation is driven by breaking bonds in the hexagonal lattice, cooperativity implies that the number of newly broken bonds is proportional to the number of already broken bonds. Because of this cooperativity, at the early stage of endocytosis bonds are broken slowly and clathrin assembly dominates over curvature generation. At the late stage of endocytosis, an increasing number of bonds are broken and curvature generation could happen rapidly and dominate over clathrin assembly. Altogether, this cooperativity leads to a constant-area-like behavior. Similar effect have been reported in Ref. (Mund et al., 2023).

Furthermore, although our model is purely energetic and does not explicitly incorporate dynamics, we observe in Figure 4(a) that along the green curve—representing the trajectory predicted by model (1,2)—the total free energy E_tot exhibits a much sharper decrease at the late stage (near the vesiculation line) compared to the early stage (near the origin). This suggests a transition from slow to fast dynamics during endocytosis. Such a transition is consistent with experimental observations, where significantly fewer number of images with large ψ_max are captured compared to those with small ψ_max (Mund et al., 2023).

Note that our model involves two distinct concepts of curvature growth. The first is the growth of imposed curvature—referred to here as intrinsic curvature and denoted by the parameter c₀—which is driven by the reorganization of bonds between clathrin molecules within the coat. The second is the growth of the actual membrane curvature, reflected by the increasing value of ψ_max. The latter process is driven by the former.

Models (1,1) and (1,2) incorporate energy terms (Equation 6) that promote the increase of intrinsic curvature c₀, which in turn drives the membrane to adopt a more curved shape (increasing ψ_max). In the absence of these energy contributions, the system faces an energy barrier separating a weakly curved membrane state (low ψ_max) from a highly curved state (high ψ_max). This barrier can be observed, for example, in the red curves of Figure 3(a–c) and in Appendix 6–Figure 1. As a result, membrane bending cannot proceed spontaneously and requires additional energy input from clathrin assembly.

The energy terms described in Equation 6 serve to eliminate this energy barrier by lowering the energy difference between the uphill and downhill regions of the energy landscape. However, these same terms also steepen the downhill slope, which may lead to overly aggressive curvature growth. To mitigate this effect, one could introduce a saturation-like energy term of the form where c_s represents a saturation curvature. Importantly, adding such a term would not alter the conclusions of our study, since the energy landscape already favors high membrane curvature (i.e., it is downward sloping) even without the additional energy terms.

When we compare the model predictions, we find that the difference in membrane morphology between models is not as big as expected, which might explain why it has been difficult to distinguish between the constant area and the constant curvature models for so long. For instance, the neck width vs. ψ_max and the invagination depth vs. ψ_max are similar for all the four models (Figure 5c, left and right). The best fitting error of the four models calculated from the geometric features are relatively close, except for the constant curvature model, which gives the worst fit (Figure 5a). The models are mainly distinguishable from the tip radius vs. ψ_max plot at the early stage of endocytosis when the membrane is nearly flat (Figure 5c, middle), i.e. for shapes with small ψ_max. However, published experimental shape at early stages of endocytosis are sparse. Our result hints that in order to distinguish between the models, collecting membrane shapes at the early stage is necessary and the relation of tip radius vs. ψ_max is the key geometric feature to tell the models apart.

In Figure 2d and e we have shown that the coat radius R_coat, which represents the projected area of the clathrin coat in the plane of the plasma membrane, as a function of ψ_max exhibit opposite trends for the constant curvature model and the constant area model. The results suggest that in experiments the projected area for the constant area model would first increase and then decrease over time, finally reaching a plateau. However, for the constant curvature model, the projected area would increase over time and finally reach a plateau without a decreasing phase. This result suggests another method to distinguish between the two models via the projected area measurement. The idea has been used in a study where the clathrin-coated pit was imaged with platinum replica and cryoelectron microscopy and tomography (Sochacki et al., 2021). The results support a constant-area-like model, consistent with the prediction of our Model(1,2), in which the dome structures have a slightly larger coating area than flat structures. On the other hand, another study has used the super-resolved live cell fluorescence imaging with TIRF to measure the growth of the clathrin coat over time (Willy et al., 2021). The authors found a smooth drop in the projected area of clathrin coat over time. However, based on a computer simulation of clathrin assembly, they concluded that the smooth drop of the projected area was the result of a constant-curvature-like model because a constant-area-like model would produce a sharp drop. We attribute the difference between their model and our model to the fact that they model the clathrin coat as a discrete lattice while we use a continuum mechanics method. More importantly, in their model, the moment at which curvature generation occurs was arbitrarily imposed at 80% of clathrin triskelions. If the transition were chosen to occur with fewer triskelions, e.g. 40%, the sharp drop in the project area might not happen in the constant-area-like model. Furthermore, the authors used the number of triskelions to monitor the progress of endocytosis which terminates when the triskelions reach the maximum number. This choice might bias towards the constant-curvature-like model because the vesiculation may not happen at all when the clathrin numbers reaches its maximum.

Comparison of our model to experimental data demonstrates that the relative bending rigidities of the coat and the membrane are constrained. Indeed, if κ_coat /κ_bare < 50, the model predicts an abrupt change (or a gap) in ψ_max at the end of vesiculation (See Appendix 2—Figure 1), i.e., a snap-through transition (Hassinger et al., 2017). If a gap in ψ_max existed, we would expect that the distribution of experimental shapes to be discontinuous, with no or very few data corresponding to a certain range of ψ. However, in the experiments (Avinoam et al., 2015), the endocytic pits shapes are continuously distributed across the ψ_max spectrum, indicating the ergodicity of ψ_max during the endocytic process. Our calculation suggests that the clathrin coat is about 50 times stiffer than the membrane.

The purpose of the clathrin-mediated endocytosis studied in our work is the recycling of membrane and membrane-protein, and the cellular uptake of small molecules from the environment — molecules that are sufficiently small to bind to the membrane or be encapsulated within a vesicle. In contrast, the uptake of larger particles typically involves membrane wrapping driven by adhesion between the membrane and the particle, a process that has also been studied previously (Góźdź, 2007; Bahrami et al., 2016). In our model, membrane bending is driven by clathrin assembly, which induces curvature. In particle wrapping, by comparison, the driving force is the adhesion between the membrane and a rigid particle. In the absence of adhesion, wrapping increases both bending and tension energies, creating an energy barrier that separates the flat membrane state from the fully wrapped state. This barrier can hinder complete wrapping, resulting in partial or no engulfment of the particle. Only when the adhesion energy is sufficiently strong can the process proceed to full wrapping. In this context, adhesion plays a role analogous to curvature generation in our model, as both serve to overcome the energy barrier. If the particle is spherical, it imposes a constant-curvature pathway during wrapping. However, the role of clathrin molecules in this process remains unclear and will be the subject of future investigation.