extracted_captions/hassinger2017design.json

{"CAPTION FIG1.png": "'Figure 1: Schematic depiction of the main mechanical steps in CME. A multicomponent protein coat forms on the plasma membrane and causes the membrane to bend inward, forming a shallow pit. As the coat matures, the membrane becomes deeply irradiated to form an open, U-shaped pit before constricting to form a closed, \\\\(\\\\Omega\\\\)-shaped bud. The bud subsequently undergoes incision to form an internalized vesicle, and the coat is recycled. Actin polymerization is thought to provide a force, \\\\(\\\\mathbf{f}_{\\\\text{s}}\\\\) to facilitate these morphological changes, particularly at high membrane tensions [5]. Our study is focused on understanding the impact of membrane tension on the morphological changes affected by the coat and actin polymerization, as indicated by the dashed box.\\n\\n'", "CAPTION FIG2.png": "'\\nFig. 2: Membrane tension inhibits the ability of curvature generating costs to induce budding. (**a**) Profile views of membrane morphologies generated by simulations in which the area of a curvature-generating coat progressively increases, covering more of the bare membrane. The curvature-generating capability, or spontaneous curvature, of the coat is set at \\\\(C_{0}=0.02\\\\,\\\\mathrm{r}\\\\mathrm{m}^{-1}\\\\), corresponding to a preferred radius of curvature of \\\\(50\\\\,\\\\mathrm{n}\\\\mathrm{m}\\\\) (12, (**a**, _Upper_**) High membrane tension, \\\\(\\\\lambda_{3}=0.2\\\\,\\\\mathrm{p}\\\\mathrm{N}\\\\mathrm{/}\\\\mathrm{n}\\\\mathrm{m}\\\\). The membrane remains nearly flat as the area of the coat increases. (**a**, _Lower_**) Low membrane tension, \\\\(\\\\lambda_{3}=0.02\\\\,\\\\mathrm{p}\\\\mathrm{N}\\\\mathrm{/}\\\\mathrm{n}\\\\mathrm{m}\\\\). Addition of coat produces a smooth evolution from a flat membrane to a closed bud. (**b**) Membrane profiles for simulations with a constant coat area in which the spontaneous curvature of the coat progressively increases. The area of the coat is \\\\(A_{\\\\mathrm{cat}}=20.106\\\\,\\\\mathrm{n}\\\\mathrm{m}^{2}\\\\). (**b**, _Upper_**) High membrane tension, \\\\(\\\\lambda_{0}=0.2\\\\,\\\\mathrm{p}\\\\mathrm{N}\\\\mathrm{/}\\\\mathrm{n}\\\\mathrm{m}\\\\). The membrane remains nearly flat with increasing spontaneous curvature. (**b**, _Lower_**) Low membrane tension, \\\\(\\\\lambda_{0}=0.002\\\\,\\\\mathrm{p}\\\\mathrm{N}\\\\mathrm{/}\\\\mathrm{n}\\\\mathrm{m}\\\\). Increasing the spontaneous curvature of the coat induces a smooth evolution from a flat membrane to a closed bud.\\n\\n'", "CAPTION FIG3.png": "'Figure 3: A snap-through instability exists at intermediate, physiologically relevant (S4), membrane tensions, \\\\(\\\\lambda_{6}=0.02\\\\,\\\\mathrm{n}\\\\mathrm{M}/\\\\mathrm{r}\\\\mathrm{n}\\\\). (A) Membrane profiles showing bud morphology before (dashed line, \\\\(\\\\mathcal{A}_{\\\\mathrm{coat}}=20,0.65\\\\,\\\\mathrm{n}\\\\mathrm{m}^{2}\\\\)) and after (solid line, \\\\(\\\\mathcal{A}_{\\\\mathrm{coat}}=20,105\\\\,\\\\mathrm{r}\\\\mathrm{m}^{2}\\\\)) addition of a small amount of area to the coat, \\\\(C_{\\\\mathrm{q}}=0.02\\\\,\\\\mathrm{r}\\\\mathrm{m}^{-1}\\\\). (B) Mean curvature at the tip of the bud as a function of the coat area. There are two stable branches of solutions of the equilibrium membrane shape equations. The lower branch consists of open, U-shaped buds, whereas the upper branch consists of closed, G-shaped buds. The dashed portion of the curve indicates \u201cunstable\u201d solutions that are not accessible by simply increasing and decreasing the area of the coat. The marked positions on the curve denote the membrane profiles shown in \\\\(A\\\\). The transition between these two shapes is a snap-through instability, in which the bud maps closed upon a small addition to area of the coat. (C) Bud morphologies before (dashed line) and after (solid line) a snap-through instability with increasing spontaneous curvature, \\\\(\\\\mathcal{A}_{\\\\mathrm{coat}}=20,106\\\\,\\\\mathrm{r}\\\\mathrm{m}^{2}\\\\), \\\\(C_{0}=0.02\\\\,\\\\mathrm{r}\\\\mathrm{m}^{2}\\\\). (D) Mean curvature at the tip of the bud as a function of the spontaneous curvature of the coat. (E) Bud morphology before (dashed line) and after (solid line) a snap-through instability with decreasing membrane tension, \\\\(\\\\mathcal{A}_{\\\\mathrm{coat}}=20,106\\\\,\\\\mathrm{r}\\\\mathrm{m}^{2}\\\\), \\\\(C_{0}=0.02\\\\,\\\\mathrm{r}\\\\mathrm{m}^{2}\\\\), \\\\(\\\\lambda_{6}=0.02\\\\,\\\\mathrm{r}\\\\mathrm{n}\\\\mathrm{M}/\\\\mathrm{r}\\\\mathrm{n}\\\\). (F) Mean curvature at the tip of the bud as a function of the membrane-fension.\\n\\n'", "CAPTION FIG4.png": "'\\nFig. 4: Bud morphology depends on bending rigidity, membrane tension, spontaneous curvature, and coat area. (A) Coat spontaneous curvature (\\\\(\\\\zeta_{0}\\\\)) vs. membrane tension (\\\\(\\\\lambda_{0}\\\\)) phase diagram. The regions of the diagram are color coded according to the final shape of the membrane for coat \u201cgrowing\u201d simulations performed with the specified values for edge membrane tension and coat spontaneous curvature. Blue denotes closed, \\\\(\\\\Omega\\\\)-buds; red denotes open, U-shaped pits; and green are situations in which closed buds are obtained via a snap-through transition. The snap-through solutions cluster about the dashed line, \\\\(\\\\text{Ves}=1\\\\), which separates the high and low membrane tension regimes (for details, see the Instability _Evists over a Range of Membrane Tensions_, Coat Areas, and Spontaneous Curvature). The lines labeled B and C, respectively, indicate the phase diagrams at right. (B) Coat area vs. membrane tension phase diagram, \\\\(\\\\zeta_{0}=0.02\\\\) nm\\\\({}^{-1}\\\\). Blue denotes closed buds, red denotes open buds, and green denotes parameters that have both open and closed bud solutions. The dashed line, \\\\(\\\\text{Ves}=1\\\\), marks the transition from low to high membrane tension. The solid line represents the theoretical area of a sphere that minimizes the Heifrich energy at the specified membrane tension (_SI Appendix, 3. Radius of a Vesicle from Energy Minimization_). (C) Coat area vs. spontaneous curvature phase diagram, \\\\(\\\\lambda_{0}=0.02\\\\) pN/nm. The dashed line, \\\\(\\\\text{Ves}=1\\\\), marks the transition between spontaneous curvatures that are capable and incapable of overcoming the membrane tension to form a closed bud. The solid line represents the theoretical area of a sphere that minimizes the Heifrich energy at the specified spontaneous curvature (_SI Appendix, 3. Radius of a Vesicle from Energy Minimization_).\\n\\n'", "CAPTION FIG5.png": "'Figure 5: The snap-through instability at physiological tension, \\\\(\\\\lambda_{0}=0.02\\\\,\\\\mathrm{pN/nm}\\\\), is abolished when the bending rigidity of the coat is increased relative to the bare membrane, \\\\(\\\\kappa_{\\\\mathrm{bare}}=320\\\\,\\\\mathrm{pN}\\\\cdot\\\\mathrm{nm}\\\\), \\\\(\\\\kappa_{\\\\mathrm{coatt}}=2400\\\\,\\\\mathrm{pN}\\\\cdot\\\\mathrm{nm}\\\\). (\\\\(A\\\\)) Membrane profiles showing a smooth progression of bud morphologies as the area at the coat is increased (\\\\(\\\\Delta_{\\\\mathrm{cat}}=10,000\\\\,\\\\mathrm{nm}^{2}\\\\), \\\\(20,000\\\\,\\\\mathrm{nm}^{2}\\\\), \\\\(28,000\\\\,\\\\mathrm{nm}^{2}\\\\)), \\\\(\\\\xi_{0}=0.02\\\\,\\\\mathrm{nm}^{-1}\\\\). (\\\\(B\\\\)) Mean curvature at the bud tip as a function of the area of the coat. The marked positions denote the membrane profiles shown in \\\\(A\\\\). There is new only a single branch of solutions (compared with Fig. 3B), indicating a smooth evolution from a flat membrane to a closed bud. (\\\\(C\\\\)) Membrane profiles showing a smooth progression of bud morphologies as spontaneous curvature of the coat is increased (\\\\(C_{0}=0.01\\\\,\\\\mathrm{nm}^{-1},0.02\\\\,\\\\mathrm{nm}^{-1},0.024\\\\,\\\\mathrm{nm}^{-1}\\\\)), \\\\(\\\\Delta_{\\\\mathrm{coatt}}=20,106\\\\,\\\\mathrm{nm}^{2}\\\\). (\\\\(D\\\\)) Mean curvature at the bud tip as a function of the spontaneous curvature of the coat showing a single branch of solutions (compare with Fig. 3D).\\n\\n'", "CAPTION FIG6.png": "'Figure 6: A force from actin assembly can mediate the transition from a U- to \\\\(\\\\Omega\\\\)-shaped bud, avoiding the instability at intermediate membrane tension, \\\\(\\\\lambda_{0}=0.02\\\\) pN/rem. Two orientations of the actin force were chosen based on experimental evidence from yeast [31] and mammalian [45] cells. (A) Schematic depicting actin polymerization in a ring at the base of the pit with the network attached to the coat, causing a net inward force on the bud. (B) At constant coat area, \\\\(A_{\\\\text{can}}=17,593\\\\) nm\\\\({}^{2}\\\\), and spontaneous curvature, \\\\(C_{0}=0.02\\\\) nm\\\\({}^{-1}\\\\), a force (red dash) adjacent to the coat drives the shape transition from a U-shaped (dashed line) to \\\\(\\\\Omega\\\\)-shaped bud (solid line). The force intensity was homogeneously applied to the entire coat, and the force intensity at the base of the pit was set such that the total force on the membrane integrates to zero. The final applied inward force on the bud was \\\\(\\\\mathbf{f}=15\\\\) pN, well within the capability of a polymerizing actin network [60]. (C) Schematic depicting actin assembly in a collar at the base, directly providing a constricting force [45]. (D) A constricting force (red dash) localized to the coat drives the shape transition from a U-shaped (dashed line) to \\\\(\\\\Omega\\\\)-shaped bud (solid line), \\\\(A_{\\\\text{coat}}=17,593\\\\) nm\\\\({}^{2}\\\\), \\\\(C_{0}=0.02\\\\) nm\\\\({}^{-1}\\\\). The force intensity was homogeneously applied perpendicular to the membrane to an area of 5,027 nm\\\\({}^{2}\\\\) immediately adjacent to the coated region. The final applied force on the membrane was \\\\(\\\\mathbf{f}<1\\\\) pN.\\n\\n'", "CAPTION FIG7.png": "'Figure 7: A combination of increased coat rigidity and force from actin polymerization ensures robust vesiculation, even at high membrane tension, \\\\(A_{0}=0.2\\\\) pN/nm, \\\\(C_{0}=0.02\\\\) nm\\\\({}^{-1}\\\\). (A) Application of the inward directed actin force (as in Fig. 6a) induces tubulation, but not vesiculation, at high tension. (B) Increasing the stiffness of the coat alone is insufficient to overcome high membrane tension (dashed line). However, increasing the coat stiffness enables the applied force to induce vesiculation and decreases the magnitude of the force required by a factor of 3. (C) Application of the constricting actin force (as in Fig. 6C) is sufficient to induce vesiculation, even at high tension. The magnitude of the applied force required is likely unrealistically high in a biologically relevant setting. (D) Increasing the coat stiffness decreases the force required to induce vesiculation by an order of magnitude.\\n\\n'", "CAPTION FIG8.png": "'Figure 8: Design principles for robust vesiculation. The rigidity of the plasma membrane, as well as the membrane tension, resists budding by curvature-generating coats. In the low tension regime, as defined by the vesiculation number, increasing the coat area or spontaneously curvature is sufficient to induce a smooth evolution from a flat membrane to a closed bud. A combination of increased coat rigidity and force from actin polymerization is necessary to ensure robust vesiculation in the high memoranderation regime.\\n\\n'", "CAPTION FIGMOVIES1.png": "'\\n\\n**Movie 51**.: Membrane tension membrane budding in which the area of a curvature-generating coat progressively increases, covering more of the bare membrane. The curvature-generating capability, or spontaneous curvature, of the coat is set at \\\\(C_{3}=0.02\\\\,\\\\mathrm{nm}^{-1}\\\\), corresponding to a preferred radius of curvature of \\\\(50\\\\,\\\\mathrm{nm}\\\\) (\\\\(12\\\\), \\\\(45\\\\)). (_Left_) High membrane tension, \\\\(\\\\lambda_{0}=0.2\\\\,\\\\mathrm{pN/nm}\\\\). The membrane remains nearly flat as the area of the coat increases. (_Right_) Low membrane tension, \\\\(\\\\lambda_{3}=0.002\\\\,\\\\mathrm{pN/nm}\\\\). Addition of coat produces a smooth evolution from a flat membrane to a closed bud.\\n\\n'", "CAPTION FIGMOVIES2.png": "'\\n\\n**Movie 52**.: Membrane tension inhibits membrane budding in which the spontaneous curvature of a curvature-generating coat progressively increases at fixed area, \\\\(A_{\\\\text{last}}=20,106\\\\,\\\\text{nm}^{2}\\\\). (_Left_) High membrane tension, \\\\(\\\\lambda_{0}=0.2\\\\,\\\\text{pN}/nm\\\\). The membrane remains nearly flat with increasing spontaneous curvature. (_Right_) Low membrane tension, \\\\(\\\\lambda_{4}=0.002\\\\,\\\\text{pN}/nm\\\\). Increasing the spontaneous curvature of the coat induces a smooth evolution from a flat membrane to a closed bud.\\n\\n'", "CAPTION FIGMOVIES3.png": "'\\n\\n**Movie 53**.: A snap-through instability occurs with increasing coat area at intermediate, physiologically relevant (54), membrane tensions, \\\\(\\\\lambda_{0}=0.02\\\\) pH/_nm_. (Left) Membrane profiles showing bud morphology as the area to the coat increases, \\\\(\\\\mathcal{C}_{0}=0.02\\\\) nm\\\\({}^{-1}\\\\). (Right) Mean curvature at the tip of the bud as a function of the coat area. There are two stable branches of solutions of the equilibrium membrane shape equations. The marked position on the curve denotes the membrane profile shown on the left.\\n\\n'", "CAPTION FIGMOVIES4.png": "'\\n\\n**Movie 54**.: A snap-through instability occurs with increasing coat spontaneous curvature at intermediate, physiologically relevant (54), membrane tensions, \\\\(\\\\lambda_{0}=0.02\\\\) pN/nm. (_Left_) Bud morphology with increasing coat spontaneous curvature, \\\\(A_{\\\\text{last}}=20,106\\\\) nm\\\\({}^{2}\\\\). (_Right_) Mean curvature at the tip of the bud as a function of the spontaneous curvature of the coat. There are two stable branches of solutions of the equilibrium membrane shape equations. The marked position on the curve denotes the membrane profile shown on the left.\\n\\n'", "CAPTION FIGMOVIES5.png": "'\\n\\n**Movie 55**.: A snap-through instability occurs with decreasing membrane tension at intermediate, physiologically relevant [54], membrane tensions, \\\\(\\\\lambda_{b}=0.02\\\\) pN/nm. (_Left_) Bud morphology with decreasing membrane tension, \\\\(A_{\\\\text{test}}=20,106\\\\) nm\\\\({}^{2}\\\\), \\\\(\\\\text{C}_{b}=0.02\\\\) nm\\\\({}^{2}\\\\). (_Right_) Mean curvature at the tip of the bud as a function of the membrane tension. There are two stable branches of solutions of the equilibrium membrane shape equations. The marked position on the curve denotes the membrane profile shown on the left.\\n\\n'", "CAPTION FIGMOVIES6.png": "'\\n\\n**Movie 56**.: The snap-through instability with increasing coat area at physiological tension, \\\\(\\\\lambda_{b}=0.02\\\\,\\\\mathrm{pH/nm}\\\\), is abolished when the bending rigidity of the coat is increased relative to the bare membrane, \\\\(k_{\\\\mathrm{bare}}=320\\\\,\\\\mathrm{pH}\\\\cdot\\\\mathrm{nm}\\\\), \\\\(k_{\\\\mathrm{coac}}=2400\\\\,\\\\mathrm{pH}\\\\cdot\\\\mathrm{nm}\\\\). (_Left_/_f_) Membrane profiles showing a smooth progression of bud morphologies as the area of the coat is increased, \\\\(C_{\\\\mathrm{A}}=0.02\\\\,\\\\mathrm{nm}^{-1}\\\\). (_Right_) Mean curvature at the bud tip as a function of the area of the coat. The marked position denotes the membrane profile shown at left.\\n\\n'", "CAPTION FIGMOVIES7.png": "'\\n\\n**Movie 57**.: The snap-through instability with increasing cost spontaneous curvature at physiological tension, \\\\(\\\\lambda_{b}=0.02\\\\,\\\\text{pH}/\\\\text{nm}\\\\), is abolished when the bending rigidity of the coat is increased relative to the bare membrane, \\\\(k_{\\\\text{base}}=320\\\\,\\\\text{pH}\\\\cdot\\\\text{nm}\\\\), \\\\(k_{\\\\text{cast}}=2400\\\\,\\\\text{pH}\\\\cdot\\\\text{nm}\\\\). (_Left_) Membrane profiles showing a smooth progression of bud morphologies as spontaneous curvature of the coat is increased, \\\\(A_{\\\\text{cast}}=20\\\\), \\\\(106\\\\,\\\\text{nm}^{2}\\\\). (_Right_) Mean curvature at the bud tip as a function of the spontaneous curvature of the coat. The marked position denotes the membrane profile shown at left.\\n\\n'", "CAPTION FIGMOVIES8.png": "'\\n\\n**Movie S5.** A force from actin assembly can mediate the transition from a U- to \\\\(\\\\Omega\\\\)-shaped bud, avoiding the instability at intermediate membrane tension, \\\\(\\\\lambda_{b}=0.02\\\\) mN/nm. (_Left_) at constant coat area, \\\\(A_{\\\\text{coat}}=17,593\\\\) nm\\\\({}^{2}\\\\), and spontaneous curvature, \\\\(C_{b}=0.02\\\\) nm\\\\({}^{-1}\\\\), a force (red dash) localized to the coat drives the shape transition from a U-shaped to \\\\(\\\\Omega\\\\)-shaped bud. The applied force on the membrane remains below \\\\(\\\\mathbf{f}=20\\\\) pN, well within the capability of a polymerizing actin network (_Sol_). (_Right_) A constricting force (red dash) adjacent to the coat drives the shape transition from a U-shaped to \\\\(\\\\Omega\\\\)-shaped bud, \\\\(A_{\\\\text{coat}}=17,593\\\\) nm\\\\({}^{2}\\\\), \\\\(C_{b}=0.02\\\\) nm\\\\({}^{-1}\\\\). The force intensity was homogeneously applied perpendicular to the membrane to an area of \\\\(5,027\\\\) nm\\\\({}^{2}\\\\) immediately adjacent to the coated region. The applied force on the membrane remains below \\\\(\\\\mathbf{f}=1\\\\) pH.\\n\\n'", "CAPTION FIGMOVIES9.png": "'\\n\\n**Movie 59.** Comparison of actin ring forces as implemented in this study and by Walani et al. (42), \\\\(\\\\lambda_{6}=0.2\\\\) pN/nm. (_Left_) Membrane profiles showing morphology during tubulation with constant inner and outer radii (\\\\(\\\\dot{n}_{\\\\rm inner}=100\\\\) nm and \\\\(R_{\\\\rm outer}=150\\\\) nm) for the ring of upward directed force at the base of the invagination. Force initially increases with increasing depth of the invagination but remains constant after tubulation. (_Right_) Membrane during tubulation. Rather than the actin force ring being fixed with inner and outer radii \\\\(R_{\\\\rm core}=100\\\\) nm and \\\\(R_{\\\\rm outer}=150\\\\) nm, it tracks the same area of the membrane (between \\\\(A_{\\\\rm outer}=31,416\\\\) nm\\\\({}^{2}\\\\)\\\\(A_{\\\\rm outer}=70,686\\\\) nm\\\\({}^{2}\\\\)) as it is pulled into the tube. We believe that the counterforce in Walani et al. (42) was implemented in this manner. Force initially increases with increasing depth of the invagination, decreases upon tubulation, and subsequently increases as the length of the tube increases.\\n\\n'", "CAPTION FIGS1.png": "'Figure S1: Schematic of the axisymmetric geometry adopted for the simulations as described in Section 1.3 The boundary conditions at the tip of the bud and the boundary of the patch were implemented as indicated. The optional boundary conditions (\\\\(\\\\lx@sectionsign\\\\)30) and (\\\\(\\\\lx@sectionsign\\\\)31) were used to obtain the value of applied force in actin-mediated inward-directed force (Figure S2) and constriction force (Figure S3) simulations, respectively (see Section 1.3.3.\\n\\n'", "CAPTION FIGS2.png": "'Figure S2: A hyperbolic tangent functional form was used to implement heterogeneous membrane properties. As an example, \\\\(y=\\\\frac{1}{2}\\\\left[\\\\tanh(\\\\gamma(x-3))-\\\\tanh(\\\\gamma(x-7))\\\\right]\\\\) is plotted with \\\\(\\\\gamma=20\\\\). The sharp transitions were ideal for specifying the boundaries of the coated region or regions of applied force, and the smoothness of the tanh function allowed for straightforward implementation into the numerical scheme.\\n\\n'", "CAPTION FIGS3.png": "'Figure S3: High membrane tension, \\\\(\\\\lambda_{0}=0.2\\\\,\\\\)pN/nm, \\\\(C_{0}=0.02\\\\,\\\\)nm\\\\({}^{-1}\\\\). At high membrane tension, the coat can grow arbitrarily large without causing a substantial deformation of the membrane. **(A)**\\\\(A_{\\\\rm cont}=251\\\\),\\\\(327\\\\,\\\\)nm\\\\({}^{2}\\\\), **(B)**\\\\(A_{\\\\rm cont}=1\\\\),\\\\(256\\\\),\\\\(637\\\\,\\\\)nm\\\\({}^{2}\\\\), **(C)**\\\\(A_{\\\\rm coat}=2\\\\),\\\\(513\\\\),\\\\(274\\\\,\\\\)nm\\\\({}^{2}\\\\)\\n\\n'", "CAPTION FIGS4.png": "'Figure S4: The size of the membrane patch has essentially no effect on the observed deformations of the membrane as long as it is sufficiently large. **(A)** Membrane profiles for identical coat areas and differing total patch areas. \\\\(\\\\lambda_{0}=0.002\\\\,\\\\)pN/nm, \\\\(A_{\\\\rm cost}=25,133\\\\,\\\\)nm\\\\({}^{2}\\\\), \\\\(C_{0}=0.02\\\\,\\\\)nm\\\\({}^{-1}\\\\). The deformations are identical for very large membrane patches. **(B-D)** Z-position of the bud tip, mean curvature of the bud tip, and energy to deform the membrane, respectively, as a function of the dimensionless area of the membrane patch. The deformation of the membrane is sensitive to small membrane patches, but is essentially identical beyond \\\\(\\\\alpha_{\\\\rm max}\\\\approx 200\\\\), particularly in terms of the tip mean curvature and the deformation energy.\\n\\n'", "CAPTION FIGS6.png": "'Figure S6: Mean curvature at the bud tip as a function of the area of the coat for the three different membrane tension cases. High membrane tension, \\\\(\\\\lambda_{0}=0.2\\\\,\\\\)pN/nm: The mean curvature at the bud tip drops to nearly \\\\(0\\\\,\\\\)nm\\\\({}^{-1}\\\\) as the size of the coat increases and the membrane stays essentially flat at the center of the patch (Figure 11-C). Low membrane tension, \\\\(\\\\lambda_{0}=0.002\\\\,\\\\)pN/nm: The mean curvature at the bud tip remains at approximately \\\\(0.02\\\\,\\\\)nm\\\\({}^{-1}\\\\) as the size of the coat increases and the membrane adopts the spontaneous curvature of the coat (Figure 12-F). Intermediate membrane tension, \\\\(\\\\lambda_{0}=0.002\\\\,\\\\)pN/nm: Reproduced from Figure 13. The mean curvature at the bud tip is lower for open buds (lower solution branch) relative to the low tension case, indicating that tension is inhibiting curvature generation by the coat. In contrast, the curvature is higher in the closed buds (upper solution branch) relative to closed buds in the low tension case, showing that membrane tension serves to shrink the size (and hence increase the curvature) of closed buds.\\n\\n'", "CAPTION FIGS7.png": "'Figure S7: Comparison of actin ring forces as implemented in this study and by Walani et al. [19], \\\\(\\\\lambda_{0}=0.2\\\\,\\\\)pN/nm. **(A)** Membrane profiles showing morphology before (dashed line, \\\\(Z_{p}=-100\\\\,\\\\)nm) and after (solid line, \\\\(Z_{p}=-500\\\\,\\\\)nm) tubulation with constant inner and outer radii (\\\\(R_{\\\\mathrm{inner}}=100\\\\,\\\\)nm and \\\\(R_{\\\\mathrm{outer}}=150\\\\,\\\\)nm) for the ring of upward directed force at the base of the invagination. This implementation of the actin force was used in Figures S9 and A&B. **(B)** Applied force as a function of the tip displacement. Force initially increases with increasing depth of the invagination, but remains constant after tubulation. This behavior is also observed in the absence of a ring force [19]. **(C)** Membrane morphology before (dashed line, \\\\(Z_{p}=-100\\\\,\\\\)nm) and after (solid line, \\\\(Z_{p}=-500\\\\,\\\\)nm) tubulation. The ring force is applied such that it is always applied between \\\\(A_{\\\\mathrm{inner}}=31,416\\\\,\\\\)nm\\\\({}^{2}\\\\)\\\\(A_{\\\\mathrm{outer}}=70,686\\\\,\\\\)nm\\\\({}^{2}\\\\) in the area-dependent parametrization (Section I.4). Rather than being fixed at a ring with inner and outer radii\\\\(R_{\\\\mathrm{inner}}=100\\\\,\\\\)nm and \\\\(R_{\\\\mathrm{outer}}=150\\\\,\\\\)nm, the ring tracks the same area of the membrane as it is pulled into the tube. The net effect of this is to limit the amount of membrane that can enter the tube. We believe that the counter force in Walani et al. [19] was implemented in this manner. **(D)** Applied force as a function of the tip displacement. Force initially increases with increasing depth of the invagination, decreases upon tubulation, and subsequently increases as the length of the tube increases, leading to a snapthrough instability with increasing force, as found by Walani et al. This instability is occurs independent of the membrane tension or the presence of a curvature-generating coat (data not shown). The physiological relevance of this implementation of the actin force and the snapthrough instability that results is unclear for two reasons: 1) It is unclear whether actin or other proteins at the base of the invagination would limit the ability of the membrane to flow into the invagination as implicitly assumed by this implementation, and 2) Because force from actin polymerization is a reaction force established through a Brownian ratchet mechanism, it is not clear that the snapthrough would occur, though the increase in the stress in the bilayer as the tube extends certainly could cause rupture, as hypothesized by Walani et al.\\n\\n'", "CAPTION FIGS8.png": "'Figure S8: Effect of Gaussian modulus variation on membrane budding via increasing coat area, \\\\(\\\\lambda_{0}=0.2\\\\,\\\\mathrm{pN/nm}\\\\). We define \\\\(\\\\Delta\\\\kappa_{G}\\\\equiv\\\\left(\\\\kappa_{G,\\\\mathrm{coat}}-\\\\kappa_{G,\\\\mathrm{bare}} \\\\right)/\\\\kappa_{0}\\\\) as the difference in the Gaussian modulus between the coated and bare membrane normalized by the bending modulus. **(A)** Membrane morphology in which there is no difference between the Gaussian modulus in the coated region and in the bare membrane, \\\\(A_{\\\\mathrm{coat}}=23,859\\\\,\\\\mathrm{nm}^{2}\\\\). The boundary between the coated region and the bare membrane lies at the bud neck. **(B)** Membrane morphology in which the Gaussian modulus is lower in the coated region than in the bare membrane, \\\\(A_{\\\\mathrm{coat}}=21,278\\\\,\\\\mathrm{nm}^{2}\\\\). The boundary between the coat and the bare membrane does not reach the bud neck, even for a vanishing neck radius. This is in agreement with the findings of Julicher and Lipowsky **[**20**]****. **(C)** Membrane morphology in which the Gaussian modulus is higher in the coated region than in the bare membrane (dashed line, \\\\(A_{\\\\mathrm{coat}}=21,278\\\\,\\\\mathrm{nm}^{2}\\\\); solid line, \\\\(A_{\\\\mathrm{coat}}=57,733\\\\,\\\\mathrm{nm}^{2}\\\\)). Note the difference in the axes as compared to (A) and (B). Closed buds do not form for typical ranges of coat area, though a deep U-shaped bud was present at the end of the solution branch (see (D)). This indicates that an additional solution branch of closed bud solutions may exist, but we were unable to obtain any closed bud solutions by our numerical scheme (see SOM Section **E**). **(D)** Mean curvature at the bud tip as a function of coat area for several values of \\\\(\\\\Delta\\\\kappa_{G}\\\\). The marked positions indicate the membrane profiles shown in (A), (B), and (C). Increasing the Gaussian modulus of the coated region relative to the bare membrane inhibits bud formation at typical coat areas, whereas decreasing the coat Gaussian modulus relative to the bare membrane can abolish the snapthrough transition otherwise present at intermediate membrane tension.\\n\\n'", "CAPTION TAB1.png": "'\\n\\n**Table 1.** Notation used in the model'", "CAPTION TAB2.png": "'\\n\\n**Table 2.** Parameter used in the model'", "CAPTION TAB3.png": "'\\n\\n**Table 3: Parameters used in the model**'"}