Coat stiffening can explain invagination of clathrin-coated membranes

Each biological cell is defined and protected by a plasma membrane, but also needs to transport nutrients or signaling molecules across it [1]. To take up extracellular material, eukaryotic cells have developed different uptake strategies [2]. Among these, clathrin-mediated endocytosis (CME) is the main uptake route and used mainly for particles in the range from 20 to 300 nm, which also includes many common viruses [3]. In order to bend the plasma membrane inwards and to form a transport vesicle, a hexagonal lattice made from the protein clathrin is assembled at the plasma membrane, which is punctuated by a few pentagons to generate curvature [4]. These networks are naturally formed by clathrin molecules, because they have the form of triskelia after assembly of three heavy chains, each decorated with one light chain. Clathrin triskelia feature multiple binding sites to each other as well as an intrinsic curvature [5]. Remarkably, clathrin triskelia can self-assemble without any additional factors into closed cages that resemble fullerenes, with twelve pentagons in a sea of hexagons [6]. In reconstitution assays with lipid vesicles, clathrin binds to the membrane through adapter proteins like AP180 and then deforms it [7]. In the cell, the situation under which clathrin lattices assemble is more complex, since here many more proteins participate in the assembly and invagination process [8]. In particular, an actin network can form around the growing pit, that can pull and push the invaginating membrane inwards [9, 10].

The exact time point at which curvature is generated during CME has been debated since many decades [11, 12]. Early work with cells supported the constant area model (CAM), in which clathrin triskelia first grow into large flat patches, which then invaginate. In contrast, reconstitution experiments with lipids and clathrin tend to favor the constant curvature model (CCM), in which curvature is generated right from the start of the clathrin assembly [7]. Today it is accepted that the actual process is a mixture of both scenarios, with both area and curvature growing in time during CME.

Recently different high-resolution imaging techniques have been used to gain insight into the spatiotemporal coordination of CME in mammalian cells [13, 14, 15, 16]. An early study with electron tomography (ET) has demonstrated that the curvature of the clathrin coats is not constant, but increases during invagination [13]. Using high-speed atomic force microscopy (HS-AFM) for imaging and nano-dissection, it has been found that clathrin lattices relax after cutting and thus store elastic energy [17]. With super-resolution microscopy (SRM), it was found that the curvature of the clathrin coat starts to be generated only once around half of the final clathrin coat has been assembled in flat patches and that later it saturates [16]. What all these techniques have in common is that snapshots are taken of clathrin coats that can be sorted according to the state of invagination, where usually the invagination angle is taken as a surrogate for time. However, until now these different datasets have not been compared with each other. In particular, the HS-AFM data has not yet been explicitly evaluated as a function of pseudo-time.

Although their molecular architecture suggests that individual clathrin triskelia can independently generate coat curvature, in recent years it has become increasingly clear that coat invagination arises from the collective interaction of clathrin triskelia within the clathrin lattice. In particular, it has been shown that invagination is favored by exchange of the clathrin light chains [18], that clathrin assembly is facilitated on curved membranes [19, 20, 21] and that clathrin triskelia undergo conformational switches that increase curvature [22]. Moreover, dissection of the clathrin coat using AFM has demonstrated that the assembled lattice is elastically frustrated [17]. Recent experimental studies also highlighted the role of actin during coat invagination [9]. In particular, it has been suggested that in mammalian cells, polymerizing actin could drive the invagination of the clathrin coat by pushing on its edges [10]. Moreover, it has been discussed that coat rearrangements [23] or lattice vacancies [24] could facilitate the invagination process.

To provide a simple mathematical description of coat assembly and invagination, we have recently suggested a conceptually transparent and analytically solvable kinetic model, the cooperative curvature model (CoopCM), that agrees with the growth kinetics observed with SRM [16]. The CoopCM is based on only a few simple assumptions that are motivated by the experimental observations, namely that the growing pit resembles a spherical cap, that area growth occurs mainly at the rim and that curvature is generated everywhere in the lattice, but in a cooperative manner, until a preferred value is reached. Although this phenomenological approach is effective, it does not address the underlying mechanisms. In principle, different physical mechanisms can contribute to drive coat invagination, including generation of spontaneous curvature, coat stiffening and line tension [25, 17].

In this work, we first discuss the CoopCM in more detail and show explicitly why it predicts a flat-to-curved transition at finite coat area. We then combine experimental data from different high-resolution techniques and different mammalian cell lines with the CoopCM to define a consensus pathway of CME. We next combine this consensus pathway with an energetic model for the composite of plasma membrane and clathrin coat. Within the framework of this model, we find that dynamic coat stiffening is a simple way in which invagination can be achieved. Because in the interface model invagination is driven by spontaneous curvature and stiffness penalizes deviations from this spontaneous curvature, high stiffness is required to stabilize the target state, but generates a prohibitively large energy penalty for the initially flat clathrin coat. Therefore, a gradual increase of stiffness could balance these two requirements of the system.

Previously, we showed that the kinetics of the invagination of the clathrin coat is described by the phenomenological CoopCM, which is justified as it effectively fits the invagination data of clathrin coats [16]. In the following, we briefly summarize the model and the underlying main assumptions. We also demonstrate that the CoopCM can be derived with both phenomenological and geometrical arguments.

Our first main assumption is that the clathrin coat takes the shape of a spherical cap. This assumption is supported well by electron microscopy [13] as well as by SRM [26, 16] for mammalian cell types (but not for yeast, which in addition has a high turgor pressure). A spherical cap is characterized by only two quantities, namely the invagination angle $\theta$ and the cap radius $R$ (Fig. 1A). The two most important geometrical quantities for kinetic and energetic models are the cap area $A$ and the edge length $\mathcal{E}$, which for the spherical cap read $A=2\pi R^{2}(1-\cos\theta)$ and $\mathcal{E}=2\pi R\sin\theta$, respectively. It is important to note that the formula for the area of the spherical cap includes the flat circular disk in the double limit when $\theta\rightarrow 0$ and at the same time $R\rightarrow\infty$ with the disk radius $R\sin\theta\approx R\theta$ remaining finite.

Instead of giving a fully dynamic description, we describe how the coat curvature $H=1/R$ evolves with $\theta$, which we use as a surrogate of time. The reason for this is that experimentally only snapshots of clathrin coats are taken, which, assuming that coat invagination is an irreversible process, can be ordered according to their opening angle $\theta$. Within the spherical cap assumption, curvature than is the second important variable. Experimentally it has been found that curvature changes the fastest at the beginning of invagination [16]. We therefore assume that curvature $H$ is the dominant fast dynamic variable and that coat area $A$ is controlled by an independent growth mechanism along the edge of the clathrin coat. The initial growth rate is called $\gamma$ and should be determined by the interaction of the clathrin triskelia, which self-assembly into a hexagonal lattice with pentagonal defects.

As the clathrin coat invaginates, the generation of coat curvature has to slow down eventually, because the clathrin coat approaches the final coat curvature $H_{0}=1/R_{0}$, which is set by the curvature of the clathrin triskelia, interactions with other clathrin triskelia and interactions with adaptor and accessory proteins at the membrane [20, 23]. This coat curvature $H_{0}$ is expected to be smaller than the curvature of clathrin cages $H_{\mathrm{c}}=1/R_{\mathrm{c}}$, which is determined solely by the curvature of the clathrin triskelia and the interactions between the other clathrin triskelia [5, 11]. The cage radius $R_{\mathrm{c}}$ has been measured to be in the range $32.5-50\,\mathrm{nm}$ [7, 27]. For typical pit sizes (typical membrane radius 40 nm [13] plus 15 nm accounting for the thickness of the clathrin coat and its gap to the membrane [28, 7]) the expected value for $R_{0}$ is around $55\,\mathrm{nm}$, which indeed is above the range for radii for the cages.

As the coat reaches the saturation curvature $H_{0}$, the coat curvature will stop increasing and a stable steady state of invagination emerges. Therefore, this curvature acts as a control mechanism for the invagination and effectively sets the pit size. Since the curvature saturation only sets in at the late stage of the invagination process, we assume that it is proportional to $H^{2}$. Importantly, the choice of $H^{2}$, rather than of $H$, is also justified phenomenologically, as it gives better fits [16]. Although $H$ is a high-level feature of the system, it is tempting to speculate that the functional form of $H^{2}$ also indicates that curvature saturation is a signature of cooperativity between clathrin triskelia.

Together, we now have a simple evolution equation for the coat curvature as a function of the invagination angle

With the initial condition $H(\theta=0)=0$ the solution to Eq. (1) is

Consequently, the radius of the coat is defined as the inverse of the coat curvature$R(\theta)=1/H(\theta)$, where $R_{0}=1/H_{0}$.

Besides this phenomenological derivation, we want to demonstrate that Eq. (1) can be also motivated using geometrical arguments. Earlier we have shown that the CoopCM can be formulated also as a fully dynamic description that predicts the characteristic square root dependence of curvature with time that was observed experimentally [16]. Until Section IV, we do not address time explicitly, because we first focus on the geometry of the system, which is sufficient to develop an energetic description. However, as done earlier for the fully dynamical version of the CoopCM, we now make the assumption that area growth along the edge occurs independently of coat invagination, which includes the possibility that invagination starts at finite area.

Assuming that the clathrin coat takes the shape of a spherical cap, defined by the coat radius $R$ (or its inverse the coat curvature $H$) and opening angle $\theta$, we can solve the area formula for the coat curvature. As a result we obtain $H=\sqrt{2\pi(1-\cos\theta)/A}$, which is justified as long as $A>0$. Expanding the coat curvature $H$ for a patch with finite area ($A>0$) in $\theta$ around the flat state ($\theta=0$) yields

The expansion is justified as long as $A$ changes on another (slower) time scale.

In order to obtain a differential equation for the coat curvature, we assume that the derivative of coat curvature $\mathrm{d}H/\mathrm{d}\theta$ takes the form of a polynomial in $H$. To include non-linear effects like cooperativity between clathrin triskelia we consider the equation to be of second order in $H$, $\mathrm{d}H/{\mathrm{d}\theta}=c_{0}-c_{1}H-c_{2}H^{2}$, where $c_{0},c_{1}$ and $c_{2}$ are expansion coefficients. We next use the expression for $H$ of Eq. (3) on the polynomial and compare the resulting equation with the derivative of Eq. (3) with respect to $\theta$. From the comparison of the two equations, we find that $c_{1}=0$ since the derivative of Eq. (3) does not include a linear term in $\theta$. Thus internal consistency of the theory implies that the non-linear saturation term of the CoopCM in Eq. (1) follows from the fact that coat invagination starts at a finite $A$, given by Eq. (3). We conclude that the CoopCM in fact predicts a flat-to-curved transition at finite coat area, in agreement with recent experimental observations [16]. Although the geometric motivation is consistent, we note that different microscopic models could lead to the same macroscopic description.

We still have to clarify at which coat area the invagination process starts. We therefore examine the area of the spherical cap using the expression for the coat radius $R(\theta)=1/H(\theta)$ (Eq. (2)) in the limit of a flat disk, i.e., $\theta\rightarrow 0$

We see that the CoopCM indeed automatically takes care of the double limit required to get a flat disc as initial condition. From Eq. (4) we also find a second, geometrical interpretation of $\gamma$ as the inverse of the patch radius $R_{\mathrm{T}}=1/\gamma$ at which the invagination of the flat coat starts. In the limit when the coat approaches maximum invagination, we get

Thus, the lower bound for the area of the fully invaginated state is $A=4\pi R_{0}^{2}$. From the expression, we deduce a second, geometrical interpretation of $R_{0}$, as the radius of the fully invaginated clathrin coat, which is valid in good approximation if the flat-to-curved transition occurs at finite coat area. Compared to Eq. (1), Eqs. (4-5) suggest a complementary perspective on the CoopCM: The CoopCM interpolates between the finite coat area where the flat-to-curved transition occurs (with circular disk radius $R_{T}=1/\gamma$) and the fully invaginated spherical coat area (with sphere radius $R_{0}=1/H_{0}$) in a purely geometric way, without any additional parameters.

We also note that the CCM is a limiting case of the CoopCM if $\gamma\rightarrow\infty$. The CAM is partially recovered from the CoopCM, when we demand $A(\theta=0)=A(\theta=\pi)$, which is true for $\gamma R_{0}=0.4411$. In this case, the corresponding coat area varies by around $10\%$ between $\theta=0$ and $\theta=\pi$ along the domain. These considerations suggest that the product $\gamma R_{0}$ can be taken as a measure which indicates how similar the CoopCM is with respect to the CCM ($\gamma R_{0}\gg 1$) and CAM ($\gamma R_{0}=0.4411$).

To illustrate the universal pathway of coat invagination as predicted by the CoopCM, in Fig. 1 the curvature (B), area (C) and edge length (D) of the clathrin coat are plotted, normalized to their maxima for $H_{0}=\gamma$. Due to the normalization and the fact that we here use $H_{0}=\gamma$, the shape of the curves for curvature, area and edge length are independent of the specific choice of $H_{0}$ and $\gamma$. Apart from the saturation behaviour in curvature, we see that invagination of the clathrin coat starts at a finite initial coat area and therefore also at a finite edge length. The edge length has a maximum in the vicinity of the equator.

We now use the CoopCM, in the form of Eq. (2), to fit experimental data sets of the coat curvature $H$ as a function of the invagination angle $\theta$ that have been recorded using different experimental methods and different cell lines. We use ET data [13], HS-AFM data [17], and SRM data [16]. The HS-AFM data is used as kindly provided by the authors [17].

For the ET data, we converted the measured tip radius of membrane pits as a function of $\theta$ to get the coat curvature $H(\theta)$ assuming the geometry of a spherical cap. Similarly, for the HS-AFM data we converted the measured coat area as a function of the coat radius to obtain $H(\theta)$. For the SRM data, clathrin-coated pits with negative curvature values and curvature values exceeding a threshold were excluded from the analysis (c3T3: $H>0.014\,\mathrm{nm^{-1}}$, SK-MEL-2: $H>0.016\,\mathrm{nm^{-1}}$, U2OS: $H>0.014\,\mathrm{nm^{-1}}$), similar to what was previously reported (note for U2OS: there was a typo in [16] for the threshold value), because these strongly curved pits lack endocytic marker AP2-GFP and might belong to the Golgi [16].

The different data sets, the rolling means, and the fits according to Eq. (2) are shown in Fig. 2A-E. For the CoopCM, we determine two parameters from the fit, $R_{0}=1/H_{0}$ and $\gamma$, which are both documented in Tab. 1. When we compare the results of the fits of the CoopCM to the CCM with $H=C$, where $C$ is a constant, and to the CAM with $H(\theta)=\sqrt{2\pi(1-\cos\theta)/A_{0}}$, where $A_{0}$ is a constant, we find that the CoopCM agrees best with the data for all used imaging techniques and cell lines. For the CoopCM, $\gamma R_{0}$ is between 0.67 and 0.83, while the flat-to-curved transition ($A(\theta=0)/A(\theta=\pi)$) occurs at around 35-52$\%$ of the final coat area (Tab. 1). This finding is consistent with the interpretation that the CoopCM predicts coat invagination at finite area, which is in between the CCM ($\gamma R_{0}\gg 1$) and the CAM ($\gamma R_{0}=0.4411$).

When we plot the datasets within a single panel, we find that all of them follow similar trends (Fig. 2F). But we also find characteristic differences. Most importantly, the scale of coat curvature is different for the various datasets. For SK-MEL-2 cells using ET in Fig. 2A, the curvature values reach around $0.02\,\mathrm{nm^{-1}}$, the largest observed values. For PTK2 cells using HS-AFM in Fig. 2B, we observe intermediate values of coat curvature of around $0.015\,\mathrm{nm^{-1}}$. For c3T3, SK-MEL-2 and U20S cells using SRM in Fig. 2C-E, we find the smallest curvature values of around $0.01\,\mathrm{nm^{-1}}$. Interestingly, the differences between the different cell lines using the same technique in Figs. 2C-E are smaller than the differences between different techniques using the same cell line in Figs. 2A and D. The finding can be corroborated when comparing the different values of the preferred coat radius $R_{0}$ in Tab. 1.

It is natural to assume that this systematic discrepancy is caused by how the clathrin coats are imaged in the different techniques. In the ET data of Avinoam et al. [13], the radii of the membranes invaginated by the clathrin coats are measured. We therefore expect ET to give large values of curvature. In the HS-AFM data of Tagiltsev et al. [17], the curvature is measured in unroofed cells directly at the clathrin coat. We thus expect intermediate values of coat curvatures for this techniques. In the SRM data of Mund et al. [16], the clathrin coat is labeled by polyclonal antibodies that bind to both clathrin heavy and light chains of permeabilized cells, with a potential preference to the intracellular side due to a higher density of binding sites. Therefore, we expect small values of coat curvature. These arguments explain why we observe that the values of coat curvature decrease from ET through HS-AFM to SRM (cf. Tab. 1).

Since the kinetics of clathrin coat invagination seem to be very similar for all cell lines and techniques (cf. Figs. 2A-F), we decided to pool the data by combining the values from all measurements. Following our previous reasoning and to exclude any bias in the available data, we corrected the measured data from ET by taking into account the thickness of the clathrin coat and its gap to the membrane. The corrected coat curvature is given by $H=1/(1/H_{\mathrm{pit}}+h_{\mathrm{cc}})$, where $H_{\mathrm{pit}}$ is the measured pit curvature and $h_{\mathrm{cc}}=15\,\mathrm{nm}$ is the estimated thickness correction [7]. Similarly, we corrected the measured data from SRM by taking into account the length of the antibodies. The corrected coat curvature is then given by $H=1/(1/H_{\mathrm{a}}-l_{\mathrm{ca}}$), where $H_{\mathrm{a}}$ is the measured antibody-labeled coat curvature and $l_{\mathrm{ca}}=15\,\mathrm{nm}$ is the estimated length of the clathrin antibody.

For the curvature-corrected data, the individual data sets are fitted in Fig. 2G. The scattering in the experimental data is considerable, which might be partially due to resolution limits. Moreover, since CME in cells involves many different types of proteins that are all underlying stochastic fluctuations, CME itself is highly stochastic. In addition, there are abortive events and there might be subpopulations, both of which are not taken into account here. However, we also note that the variability within single data sets is larger than the variability between the fits of the individual data sets, which seems to indicate that CME is rather similar across imaging methods and cell types. Thus, we decided to fit and evaluate the pooled data in order to find the mean trajectory for CME (cf. Fig. 2H and Tab. 1). We see good agreement between experimental data and the CoopCM and conclude that the resulting consensus pathway can be used as a reasonable basis for an energetic analysis. We also find that the flat-to-curved transition occurs at around $50\%$ of the final coat area.

In this work, we investigated the invagination pathways of clathrin coats from different angles, combining the analysis of experimental data with modelling. The analyzed data sets were acquired with very different methods, namely ET [13], HS-AFM [17] and SRM [16], but all of them are high resolution spatial data that suggest a spherical cap shape of the clathrin pits and similar changes in curvature during invagination. The spherical cap assumption is further supported by a detailed analysis of the SRM data with a maximum-likelihood model [26]. Since the spherical cap matched the shape of the majority of clathrin coats in the SRM data [16], and only some sites exhibit asymmetric and irregular deformations, we believe that the spherical cap assumption is well suited to describe the coat shape in our theoretical model.

The geometrical time courses of the invaginations seem to be fitted well across imaging techniques and cell types by the recently introduced cooperative curvature model (CoopCM) [16], as shown in Fig. 2A-F. Comparing the trajectories of the different datasets by means of the CoopCM after correcting for experimental uncertainties, we found very similar invagination behavior (Fig. 2G). We concluded that clathrin coats follow a consensus invagination pathway described by the CoopCM (Fig. 2H). We note, however, that ultimately the CoopCM is derived phenomenologically and in principle other descriptions might exist, although, to date, we know of no other suitable candidate. In particular, a linear growth model was tested before to give worse results [16]. The CoopCM suggests that the dynamics of invagination is strongly determined by cooperative generation of curvature in the clathrin lattice. In the future, this general conclusion should be validated by a more microscopic model.

One important result from our model is that a flat-to-curved transition occurs at finite coat area, which agrees with experimental observations [13, 14, 16]. While our model describes the averaged pathway of coat invagination, we cannot rule out that different populations contribute to the analyzed datasets. Subpopulations might also follow different pathways and mechanisms for coat invagination. Nevertheless, we concluded that the CoopCM defines a consensus pathway which can be used for an energetic analysis.

We then combined the consensus pathway and the CoopCM with an energetic model for the composite of membrane and coat. Our main result was that the energy related to coat stiffness dominates all other contributions. Moreover, within the framework of our interface model and assuming no further energetic contributions other than those of the membrane and the clathrin coat, our analysis suggests that the coat stiffness changes during coat invagination. When using a constant value for stiffness and taking the high values measured for pits in their late stages, our model predicts that coat invagination is energetically unfavorable. Our model therefore predicts that coat stiffness starts at low values and dynamically increases during invagination, in agreement with the emerging notion that clathrin coats are plastic and cooperative. Although there is no experimental evidence to date, we point out that additional unreported molecular mechanisms may drive clathrin coat invagination.

We showed that the CoopCM can be derived from the energetic model when assuming that the coat polymerization energy increases linearly with coat curvature and the clathrin coat stiffens with a power law exponent of three with coat curvature. We note that a power law exponent of three also relates bending stiffness and thickness of thin sheets [38], suggesting that effective thickness might grow linear with curvature. In practise, this viewpoint of continuum mechanics might however be too naive, because clathrin lattices have very specific discrete architectures. Rather, this prediction of our theory should be tested experimentally. Our prediction that the polymerization energy of the clathrin coat increases linearly with the curvature of the coat is consistent with experiments and theory predicting that clathrin assembly is stabilized on curved surfaces and that clathrin triskelia in curved coats contain more energy than in flat clathrin lattices [20, 39]. In fact, the possibility of a curvature-dependent polymerization energy has been discussed previously [31]. Possible driving factors for coat stiffening and increase of the effective thickness on a more microscopic level are accumulation of phosphorylated clathrin light chains [18], conformational changes in clathrin triskelia [22], rearrangements within the coat or mending of lattice defects [23], filling up lattice vacancies [24], release of elastic energy within the coat [17], increasing the density of the clathrin coat [40], solidification of the coat [30, 41], changing clathrin to AP2 adapter ratio during coat assembly [14], the organized binding of clathrin triskelia to adaptor clusters [42], mechanics of the coupling between membrane and clathrin coat through adapter proteins [28], and the stabilization of membrane curvature by clathrin in a ratchet-like manner [21].

All these mechanisms require flexibility and weak interactions within the coat during invagination initially. Lattice rearrangements seem plausible, given that the coat shows triskelia exchange in the flat state [43, 13] but also can disassemble fast once it is complete [8]. The notion of weak interactions is supported by the fact that the legs of clathrin triskelia are flexible and bind only weakly to each other [44]. Moreover, clathrin triskelia are not perfectly aligned with respect to each other even in curved configurations [5].

The stiffening of the clathrin coat could be complemented by an increasing line tension $\zeta$, for example by polymerizing actin pushing from the periphery of the coat [9, 10]. However, our data also shows that this might be dispensable under normal conditions, in line with previous experiments [45]. Moreover, the scenario would still require that the clathrin coat stiffens during invagination, because otherwise flat clathrin coat assembly cannot be explained. However, analyzing this scenario is beyond the scope of our model and future work will therefore be required. Our model is also not suited to analyze the invagination of clathrin coats that do not assume a spherical shape as for example in yeast [46].

Our interface model describes the mean invagination pathway of the clathrin coat and does not consider deviations from the spherical cap shape or clathrin lattices that are not regularly assembled. It will require future work and more microscopic models to dissect whether these aspects can impact the energetics of coat invagination and how coat stiffening occurs microscopically. In principle coat stiffening could happen in a continuous way, similar to releasing elastic energy within the coat [17], or in a discrete or ratchet-like way, for example due to conformational changes in the clathrin triskelia, rearrangements within the clathrin coat [22, 23], or by filling up lattice vacanices [24]. Therefore, experimentally the CoopCM could be tested by studying the thickness, the density or the mobility of single triskelia within the clathrin coat as a function of time.

The notion of coat stiffening might be related to the ongoing discussion whether clathrin can generate membrane curvature alone [47] or whether it is only stabilising membrane curvature generated by adaptors and accessory protein [21, 48]. If membrane curvature was generated mostly by adaptor proteins, clathrin could stabilize the constant membrane curvature and the clathrin coat would immediately grow in a stiff configuration. However, if membrane curvature was mostly generated by clathrin, the coat curvature would increase during assembly because the clathrin coat stiffens. We speculate that coat stiffening occurs when the ability to generate membrane curvature by other proteins is insufficient. If so, clathrin assembly and coat stiffening could then drive membrane invagination. If coat stiffening is dispensable because coat curvature is generated by other proteins, this notion could also explain the seemingly conflicting experimental findings that report that clathrin coat invagination follows a model different from the flat-to-curved model [15, 49, 50].

To conclude, our results imply that the clathrin coat is more plastic during coat invagination than formerly appreciated and dynamically stiffens during coat invagination.

We thank Markus Mund, Aline Tschanz and Jonas Ries for helpful discussions and a critical reading of the manuscript. We also kindly acknowledge Simon Scheuring for providing the HS-AFM data for the analysis of clathrin coat invagination. We thank the reviewers of previous versions of this manuscript for useful feedback that helped us to improve this work. F.F. acknowledges financial support by the NOMIS foundation. U.S.S. was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project number 240245660 (SFB 1129). Moreover he is a member of the Interdisciplinary Center for Scientific Computing (IWR) at Heidelberg.