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Conceived and designed the experiments: NJA JN RR. Performed the experiments: NJA. Analyzed the data: NJA RR. Contributed reagents/materials/analysis tools: NJA JN. Wrote the paper: NJA RR.

The authors have declared that no competing interests exist.

In eukaryotic cells, the internalization of extracellular cargo via the endocytic machinery is an important regulatory process required for many essential cellular functions. The role of cooperative protein-protein and protein-membrane interactions in the ubiquitous endocytic pathway in mammalian cells, namely the clathrin-dependent endocytosis, remains unresolved. We employ the Helfrich membrane Hamiltonian together with surface evolution methodology to address how the shapes and energetics of vesicular-bud formation in a planar membrane are stabilized by presence of the clathrin-coat assembly. Our results identify a unique dual role for the tubulating protein epsin: multiple epsins localized spatially and orientationally collectively play the role of a curvature inducing capsid; in addition, epsin serves the role of an adapter in binding the clathrin coat to the membrane. Our results also suggest an important role for the clathrin lattice, namely in the spatial- and orientational-templating of epsins. We suggest that there exists a critical size of the coat above which a vesicular bud with a constricted neck resembling a mature vesicle is stabilized. Based on the observed strong dependence of the vesicle diameter on the bending rigidity, we suggest that the variability in bending stiffness due to variations in membrane composition with cell type can explain the experimentally observed variability on the size of clathrin-coated vesicles, which typically range 50–100 nm. Our model also provides estimates for the number of epsins involved in stabilizing a coated vesicle, and without any direct fitting reproduces the experimentally observed shapes of vesicular intermediates as well as their probability distributions quantitatively, in wildtype as well as CLAP IgG injected neuronal cell experiments. We have presented a minimal mesoscale model which quantitatively explains several experimental observations on the process of vesicle nucleation induced by the clathrin-coated assembly prior to vesicle scission in clathrin dependent endocytosis.

Cell membranes and membrane-based organelles actively mediate several intracellular signaling and trafficking decisions. A growing number of applications rely on cooperative interactions between molecular assemblies and membranes. Yet, the studies of membrane-based and membrane-mediated signaling are not considered core aspects of systems biology. While a coherent and complete description of cell membrane-mediated signaling is not always possible by experimental methods, multiscale modeling and simulation approaches can provide valuable insights at microscopic and mesoscopic scales. Here, we present a quantitative model for describing how cell-membrane topologies are actively mediated and manipulated by intracellular protein assemblies. Specifically, the model describes a crucial step in the intracellular endocytic trafficking mechanisms, i.e., active transport mechanisms mediated through budding of the cell membrane orchestrated by protein-interaction networks. The proposed theory and modeling approach is expected to create avenues for many novel applications in systems biology, pharmacology, and nanobiotechnology. The particular application to endocytosis explored here can help discern pathological cellular trafficking fates of receptors implicated in a variety of biomedical conditions such as cancer, as well as impact the technology of targeted drug delivery in nanomedicine.

The cellular process of endocytosis is important in the biological regulation of trafficking in cells, as well as impacts the technology of targeted drug delivery in nanomedicine

We focus on the energetic stabilization of a budding vesicle induced by the clathrin-coat assembly. Recent work

Even though models directly addressing CDE in the experimental (cellular) context have not been proposed, Oster et al. have addressed yeast endocytosis driven by actin _{B}T derived from literature

The free energy of state 2 relative to state 1 is described by E_{t}.

Clathrin triskelia and AP-2 (in a ratio of 1∶1) polymerize to form a coat _{B}T. The inclusion of epsin in the clathrin-coat accounts for −23 k_{B}T of energy per bound epsin: the ENTH domain of epsin binds to the PtdIns(4,5)P_{2} (or PIP2) lipid head groups on the membrane with a binding energy of −14 k_{B}T per bound epsin _{B}T ^{7/4} is 29 ^{2} = 7850 nm^{2}. Considering the area per lipid head-group to be 0.65 nm^{2}, the number of PIP2 molecules in the membrane spanning the area of the coat is 1% of (7850/0.65) = 185. Hence, we note that the ratio of ENTH binding sites (which correspond to the PIP2 on membrane) to the CLAP binding sites (which correspond to the triskelia) is 185/29≈6, and hence as the clathrin coat grows, we expect sufficient number of the corresponding PIP2 binding sites to be present for the ENTH domain of epsin to bind. For this reason, we are justified in not explicitly considering PIP2 as a necessary/limiting species in our minimal model.

Field-theoretic approaches are popular for studying energetic and entropic contributions in continuum field-based mesoscale models

The full membrane profile is obtained by rotating the curve by 2π about the z-axis.

Here, H is the mean curvature of the membrane, H_{0} is the imposed (or intrinsic) curvature of the membrane due to curvature-inducing proteins and is a function of arc-length s,

Here, _{0} (or at s = s_{1}) corresponding to the pinning of the membrane by the cytoskeleton at the boundary of the membrane patch. In addition, due to the axis-symmetry, at R = 0, _{1} is not known _{0}; in this work, we employ R_{0} = 500 nm. We also compute the curvature deformation energy of the membrane defined by:

We present our results for the case when interfacial frame tension σ is zero. Results obtained for non-zero σ (not shown) are found to be similar to the σ = 0 case. We also note that in prior work, we showed that the entropic term |TΔS| at T = 300K is small, i.e. ∼5% of the membrane bending energy for κ = 20 k_{B}T _{a}(s_{0}) is computed using the relationship,_{0}) is the radius at s_{0}, which marks the coat boundary.

In our model, the dominant factor contributing to the intrinsic curvature H_{0} in the region where the membrane binds to the clathrin coat is the presence of epsins, bound at the CLAP-binding sites on the coat. In a recent study,

That is, for the nature of epsin-induced curvature, we have assumed a form that has a spatial decay. Such a choice of spatially-varying intrinsic curvature function is motivated by recent molecular simulations

In vitro, Ford et al. _{0} = 0.1 nm^{−1}. Using the surface-evolution approach, we calculate the curvature deformation energy of the membrane, E_{c} (defined in Eq. (3)) when a single epsin interacts with the membrane, i.e. through the curvature function in Eq. (5). Since the energy E_{c} is stabilized by the negative interaction energy of the ENTH domain of epsin with the membrane (E_{r}), we iteratively determine the value of b in Eq. (5) such that E_{c}≈|Er|; using E_{r} = −14k_{B}T _{B}T.

The periodicity of clathrin lattice, (from cryo-EM studies _{0} of the form:

Here, the index i runs over the number of concentric shells of epsins on the coat separated by a distance of 18.5 nm, the underlying periodicity of the clathrin lattice. Hence, relative to a central epsin bound to the coat at R = 0 and s_{0,1} = 0, successive shells of epsins are located at s_{0,2} = 18.5 nm, s_{0,3} = 37 nm, s_{0,4} = 55.5 nm, etc. until we reach the periphery of the coat of a prescribed extent (or area A_{a}); the H_{0} function is depicted in _{0} for the outermost shell and the neck-radius R(s_{0}) is the radius at this value of s_{0}, as described earlier. In _{a} (defined in Eq. (4)) obtained using the surface evolution method and subject to the epsin curvature fields described by Eq. (6); we find that above a critical value of the coat area, the membrane profile develops overhangs, (also evident from the behavior of the neck-radius in _{a} approaches 6500 nm^{2}, transforms to a mature spherical vesicular bud with a narrow neck. We emphasize the generality of this result, i.e., that there exists a critical coat area above which the membrane deformation develops an over-hang and a constricted neck, by confirming this observed trend using a conceptually simplified “capsid model” in which H_{0}(s) = 0.08 nm^{−1} if s<s_{0} and H_{0}(s) = 0 if s≥s_{0}, s_{0} is the length of the clathrin coat, as described in _{epsins},_{i} in each shell i as:

(_{B}T. For the largest coat area, the membrane shape is reminiscent of a clathrin-coated vesicle. (_{a}.

Our results for the epsin shell model assumed a value for the bending rigidity of κ = 20 k_{B}T reported in the literature _{B}T has been reported in the literature: in particular, there is consensus that cytoskeleton-free membranes have rigidity in the range of 20 k_{B}T and cytoskeleton-fortified membranes can be as stiff as 400 k_{B}T. For this reason, it has indeed been postulated that apparent bending rigidity of the membrane depends on the relevant length scale and lies between 20 k_{B}T (membrane patches below 100 nm) and 500 k_{B}T (membrane patches of 1 µm) _{B}T on the mechanism of epsin-induced vesicular bud formation. In _{epsins} and the diameter of the vesicular bud, d, were also computed as depicted in

The computed deformation energy E_{c} (defined in Eq. (3)) for the capsid model is plotted in _{a}; we find that the energy E_{c} required to form a mature spherical bud of diameter 50 nm is estimated to be 25κ = 500 k_{B}T. The estimate is very close to 8πκ, which is the deformation energy of a spherical vesicle of diameter d for which H_{0} = 4/d (and constant in space). The energy E_{c} required to deform the membrane can be offset by stabilizing interactions between the proteins in the clathrin coat assembly and between the coat proteins and the membrane. As described in the introduction, the free energy of the clathrin-coat assembly, E_{a} was estimated by Nossal _{B}T, i.e., |E_{c}|≫|E_{a}|. This implies that the curvature induction in the presence of a clathrin-coat is energetically unfavorable in the absence of additional stabilizing interactions. Indeed, as reported in cell-experiments _{r} which includes those interactions that preferentially stabilize state 2 over state 1 in _{epsin} = −23 k_{B}T per bound epsin and hence, within our model, we consider E_{r}(A_{a}) = N_{epsins}(A_{a})×ε_{epsin}. Thus, for a given extent of the coat characterized by its area A_{a}, the total free energy change of the membrane and clathrin-coat assembly in the curved state (state 2, see _{t}(A_{a}) = E_{c}(A_{a})+E_{a}(A_{a})+E_{r}(A_{a}).

Recently, Jakobsson et al.

(a) Calculated and (b) experimental probability of observing a clathrin-coated vesicular bud of given size in WT cells (filled) and CLAP IgG injected cells (unfilled). In the calculated histogram, the four categories defined are based on the progression of bud growth. Category 2 includes all buds for which bud diameter is less compared to the neck radius, while category 4 includes all buds for which bud diameter is more compared to the neck radius. Category 3 is an intermediate case between category 2 and 4.

By computing E_{c} and E_{r} for different values of A_{a} in the capsid model, we determine the energetics of the clathrin coated vesicular bud E_{t} versus coat area, A_{a} for the capsid model (see _{0}) for the capsid model is slightly different from that for the shell model. We also computed probability of observing different coated-intermediates of vesicular structures as P∝exp(−E_{t}(A_{a})/k_{B}T) as depicted in _{epsins}(CLAP cells) = N_{epsins}(WT cells)*A_{a}(vesicles in CLAP injected cells)/A_{a}(in WT cells) = 33. The ratio of the respective areas ( = 1.6) is determined based on the experimental observations of increase in the size of the coated intermediates in CLAP injected cells relative to WT cells _{epsins} = 33 and ε_{epsin} = −14 k_{B}T (reduced from −23 k_{B}T due to the abrogation of the CLAP-clathrin/AP-2 interaction), we find not only that E_{t}(A_{a}) increases monotonically with A_{a} (a reversal in trend, see _{t}(A_{a})/k_{B}T) quantitatively matches the experimentally observed distribution in CLAP IgG injected cells, (compare _{epsins} increase in the CLAP IgG injected cells relative to wildtype, the size of the bud likely increases due to a lack of templating of epsins; arguably, there is lack of bond-orientational order as the CLAP domains of epsin can no-longer bind the periodic clathrin lattice. Corroborating this view, many extended coated structures (_{r} and should make the coated vesicular bud highly unfavorable. Indeed, consistent with this view, in cells microinjected with ENTH antibodies the extent of clathrin-coated structures decreased by over 90% _{B}T which is 0.6 kcal/mol at T = 300 K) can lead to a large change in the fraction [exp(0.6)≈factor of 2]. Hence, an order of magnitude agreement in histograms between theory and experiment in the trends of the intermediate shapes implies that the energetics agree even more closely.

In conclusion, we have presented a minimal mesoscale model which we believe imposes the correct spatial as well as thermodynamic constraints, and quantitatively explains several experimental observations on the process of vesicle nucleation induced by the clathrin-coated assembly prior to vesicle scission in CDE. We re-iterate that the input to our model is the membrane bending rigidity, spacing between epsins bound to the clathrin coat, and the curvature-field imposed by each bound epsin, which have all been determined using independent biophysical experiments. For these choices of input, our calculations then yield the membrane profiles for different sizes of the clathrin coat. Based on the number of shells of epsins accommodated on the clathrin coat (which depends on the size of the coat), and the circumference of each shell (which depends on the coat/membrane deformation), the number of epsins is calculated. Thus, the number of epsins, the membrane profile, and the deformation energy are outputs of our model. While our model does not include nucleation of the clathrin coat or scission of a mature coated vesicular-bud, our results identify a unique dual role for the tubulating protein epsin: multiple epsins localized spatially and orientationally collectively play the central role of a curvature inducing capsid; in addition, epsin serves the role as an adapter in binding the clathrin coat to the membrane. Our results also suggest an important role for the clathrin lattice, namely in the spatial- and orientational-templating of epsins for providing the appropriate curvature field for vesicle budding. We suggest that there exists a critical size (area) of the coat above which a vesicular bud with a constricted neck resembling a mature vesicle is stabilized. Based on the strong dependence of the vesicle diameter on the bending rigidity, we suggest that the variability in bending stiffness due variations in membrane composition with cell type can explain the experimentally observed variability on the size of clathrin-coated vesicles, which typically range 50–100 nm.

Apart from providing a mechanistic description of the budding process in CDE, our model provides estimates for the number of epsins involved in stabilizing a coated vesicle, and without any direct fitting, reproduces the experimentally observed shapes of vesicular intermediates as well as their probability distributions quantitatively in wildtype as well as CLAP IgG injected neuronal cell experiments. We consider such an agreement to be a strong validation for the basis of our model. These model predictions can further be tested by engineering mutations in epsin, clathrin, and AP-2 all of which are predicted to influence the distribution of coated structures. The framework of our approach is generalizable to vesicle nucleation in clathrin-independent endocytosis. Indeed, based on our results we can speculate that alternative mechanisms (such as receptor clustering) which can provide a hexatic bond-orientational templating of epsins on the membrane can facilitate vesicle-bud formation independent of CDE

The capsid model. Three different membrane deformation profiles under the influence of clathrin imposed curvature for s_{0} = 25, 50 and 70 nm. For s_{0} = 70 nm, membrane shape is reminiscent of a clathrin-coated vesicle. Inset (top): A schematic of the membrane profile explaining various symbols in the surface evolution methodology. The full membrane profile is obtained by rotating the curve by 2π about the z-axis. Inset (bottom) shows spontaneous curvature function experienced by the membrane due to the clathrin coat assembly in the capsid model.

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The capsid model. Curvature deformation energy of the membrane versus the area of the clathrin coat, A_{a}(s_{0}) for different values of s_{0}: 25nm–70nm. Inset: vesicle neck-radius R(s_{0}) plotted against coat area A(s_{0}) for different values of s_{0}: 25 nm–70 nm.

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Epsin shell model. Radius R versus s in the epsin shell model.

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Epsin shell model. Determination of the range parameter b as a function of bending rigidity.

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a) A schematic (corresponding to a mature bud in

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Energetics of the clathrin coated vesicular bud E_{t} versus coat area, A_{a} for the capsid model.

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Membrane shape equations and details of the numerical scheme.

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The capsid model.

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