Rossana
Rojas Molina
^{a},
Susanne
Liese
^{a},
Haleh
Alimohamadi
^{b},
Padmini
Rangamani
^{b} and
Andreas
Carlson
*^{a}
^{a}Mechanics Division, Department of Mathematics, University of Oslo, 0316 Oslo, Norway. E-mail: acarlson@math.uio.no
^{b}Department of Mechanical and Aerospace Engineering, University of California, San Diego, CA 92093, USA

Received
2nd June 2020
, Accepted 12th October 2020

First published on 14th October 2020

A wide range of proteins are known to create shape transformations of biological membranes, where the remodelling is a coupling between the energetic costs from deforming the membrane, the recruitment of proteins that induce a local spontaneous curvature C_{0} and the diffusion of proteins along the membrane. We propose a minimal mathematical model that accounts for these processes to describe the diffuso-kinetic dynamics of membrane budding processes. By deploying numerical simulations we map out the membrane shapes, the time for vesicle formation and the vesicle size as a function of the dimensionless kinetic recruitment parameter K_{1} and the proteins sensitivity to mean curvature. We derive a time for scission that follows a power law ∼K_{1}^{−2/3}, a consequence of the interplay between the spreading of proteins by diffusion and the kinetic-limited increase of the protein density on the membrane. We also find a scaling law for the vesicle size ∼1/(_{av}C_{0}), with _{av} the average protein density in the vesicle, which is confirmed in the numerical simulations. Rescaling all the membrane profiles at the time of vesicle formation highlights that the membrane adopts a self-similar shape.

Reconstituted and synthetic vesicles are essential model systems that help to reveal the fundamental biophysical mechanisms by which proteins are able to induce membrane shape transformations.^{16–21} For instance, experiments have demonstrated that the formation of tubular structures is directly correlated with the protein density on the membrane^{16} and that protein crowding correlates with the formation and abscission of vesicles.^{17} Moreover, it has been shown experimentally that the local membrane curvature and the resulting membrane shape is coupled to the concentration of curvature-inducing macromolecules. For example, tubular structures that are formed by anchoring polymers, shrink as diffusion reduces the local polymer density.^{21}

A typical membrane remodeling process starts from a flat surface and develops into a deformed membrane with the shape of a bud, vesicle or tubule^{22,23} as proteins are locally recruited to the membrane surface and induce a spontaneous curvature.^{24,25} Membrane deformation is thus driven by a gradual recruitment and accumulation of membrane-associated proteins and changes in physical properties over time,^{8} suggesting that in order to derive a theoretical description of the dynamic evolution of a membrane shape we must include the recruitment of curvature-inducing proteins.

Over the years, numerous theoretical studies have been dedicated to describe a wide range of mechanisms that play a vital role in various cellular processes, i.e. diffusion of transmembrane proteins,^{26} protein crowding^{27–29} and spontaneous curvature induced by macromolecules such as polymers and proteins.^{18,21,30–32} More complex theoretical models also include the viscous dissipation generated as the proteins move on the lipid bilayer,^{33,34} as well as non-local hydrodynamics where the entire flow field is resolved.^{35–37}

Since the thickness of a lipid bilayer is much smaller than the typical length scale of membrane deformations, it is common to treat biomembranes as elastic thin sheets.^{38–41} Similarly, the size of an individual protein is at least an order of magnitude smaller than the extension of a membrane bud or vesicle, which justifies to describe the proteins as a continuous field. Previous theoretical studies have investigated various aspects of membrane deformation generated by concentration-dependent spontaneous curvature, considering either a static protein distribution on the membrane^{42–45} or including diffusion dynamics.^{26,30,34} In addition, a theoretical framework based on the Onsager variational principles has been developed, where the dynamics are given by the balance between dissipative and driving forces.^{33,46,47} This framework gives a compact mathematical description of adsorption and desorption of proteins from the bulk and its diffusive dynamic on a fixed membrane shape.

Compared to the diffusive protein dynamics on membranes, far less is known about the dynamic interplay between diffusion, local protein kinetics and membrane shape changes, where a theoretical model considering all three processes has not yet been explicitly established. In this study, we develop such a model with a focus on finding temporal relationships between the kinetics of protein recruitment and the timescales of bud formation.

To study the spatio-temporal budding process of a membrane, we develop a minimal mathematical model for the diffuso-kinetics of membrane associated proteins. We treat the concentration of proteins as a continuous field following a diffusion equation with an adsorption term, describing the protein recruitment from a reservoir i.e. the cytosol or the extracellular space, and a detachment term representing the protein turnover. Additionally, our model incorporates that the protein concentration induces an effective local spontaneous curvature on the membrane, which drives the membrane shape evolution. Finally, to characterize the membrane deformation dynamics we span the phase space by varying the kinetic parameters and the protein sensitivity for the detection of membrane curvature to uncover scaling relationships between time for scission and kinetic recruitment parameters of proteins onto the membrane.

(1) |

The first term is the Helfrich energy,^{26,38} where H is the mean curvature and B is the bending rigidity. The Helfrich model is suitable to describe cases where the radii of membrane curvatures are much larger than the thickness of the bilayer,^{40} allowing us to treat the lipid bilayer as a thin elastic shell. Here, we assume that the induced spontaneous curvature C = C_{0} due to membrane–protein interactions depends linearly on the protein density,^{26,48,49} where C_{0} is a proportionality constant associated to the spontaneous curvature induced by one protein and is the protein density on the membrane scaled by the saturation density.^{49} The proteins are mobile on the membrane, hence the density varies in time and space.

The second term in eqn (1) is the surface tension. We describe the membrane as an infinite surface, where the far-field acts as a lipid reservoir. In this case, a constant surface tension λ acts along the entire membrane. The third term accounts for entropic effects. When the protein density in the membrane surface is small and the available binding sites for the proteins are not bounded, the entropy term is well approximated by the mixing entropy of an ideal gas.^{46,50,51} This model is simpler than the general Langmuir absorption model,^{52,53} where k_{b} is the Boltzmann constant, T is the temperature and a_{p} is the area occupied by one protein.

Additional terms may be included in W (eqn (1)), such as interaction terms between proteins ∼^{2} and the energy cost arising from density gradients ∼(∇).^{2,48,54} Both terms scale with the magnitude of the protein–protein interaction. Since the non-specific interaction between proteins is weak compared to the bending and entropic energy, the interaction and gradient terms can be neglected. In the ESI,† (Section S8), we estimate the contribution to the energy functional by the interaction between proteins and the density gradients, to illustrate that the mixing entropy is the leading contribution to the energy, thus simplifying its description. To keep the mathematical model minimal, we assume that both the Gaussian bending modulus and the bending rigidity B are constant, i.e., they do not depend on the local protein density, which also implies that the Gaussian bending energy is a constant, since we do not consider topological changes such as membrane scission.^{30,55}

The mean curvature H in the arc-length parametrization is given by:^{56}

(2) |

The operator represents the derivative with respect to the arc-length s.

The arc-length parametrization allows to express the coordinates r, z and the area of the membrane, A, in the following way:

r′ = cosϕ | (3) |

z′ = sinϕ | (4) |

A′ = 2πr | (5) |

The shape of the membrane for a given protein distribution is such that it must minimize the total energy, given by the integral of eqn (1) over the total area of the membrane, .

To derive the energy minimizing shape, we define

(6) |

The bending moment of the membrane, M, is given by:^{58}

(7) |

From eqn (7) we obtain the differential equation for the angle ϕ as:

(8) |

Finally, following the Euler–Lagrange formalism, we obtain (see the ESI,† for details):

(9) |

(10) |

The boundary conditions implemented to solve the set of 6 equations given by eqn (3)–(5) and (8)–(10), which enforce a transition into a flat membrane at the outer boundary, are described in detail in the ESI.†

(11) |

In general, the protein flux is given in terms of the chemical potential derived from the energy functional in eqn (1), .^{46,60}Λ is the protein mobility and is the functional derivative of the energy functional W with respect to the protein density. In the absence of gradient terms in the energy, the functional derivative reduces to .^{61} Hence, the non-vanishing component of the flux, J, is given by:

(12) |

(13) |

To model the protein recruitment, we make four assumptions: first, the recruitment is modeled following the linear adsorption–diffusion model,^{46} in which it is assumed that the protein density is small, and that the available binding sites for the proteins are not bounded, as in the more general Langmuir absorption model. Second, we assume that the protein density in the bulk is constant. Third, protein recruitment is triggered when the membrane curvature exceeds a threshold value H_{0}. This assumption is inspired by experimental observations, which found that certain proteins are enriched in curved regions of the membrane^{62,63} with the ability to also induce a curvature.^{64} Theoretical studies based on molecular dynamics simulations^{65} and Monte Carlo simulations^{66} have shown that various biophysical mechanisms can cause curvature sensing, where proteins adsorb to a membrane in a step-like manner with respect to the membrane curvature. We incorporate these key characteristics in a phenomenological curvature sensing model by multiplying the on-rate by a Heaviside function Θ(H − H_{0}) and show in the ESI† (Section S7) that regularizing the Heaviside function does not affect the prediction as long as the jump is sufficiently steep. Lastly, we consider the diffuso-kinetic dynamics i.e. out equilibrium, which is further illustrated below by our numerical simulations. We acknowledge that a more complex relation between the recruitment kinetics and the membrane curvature might be proposed,^{67} which requires a more elaborated theoretical treatment also satisfying detailed balance at equilibrium. To the best of our knowledge, a universal model for curvature-sensitive recruitment dynamics is still a topical question in the field.

In light of these assumptions, the mathematical form of _{source} can be written as:

_{source} = c_{p}k_{on}Θ(H − H_{0}) − k_{off} | (14) |

Parameter | Typical value |
---|---|

Membrane thickness h | 5 nm^{68} |

Membrane viscosity η_{m} |
(10^{−9}–10^{−7}) N s m^{−1}^{33,69,70} |

Cytosol viscosity η_{c} |
(1–4) × 10^{−2} N s m^{−2}^{71,72} |

Spontaneous curvature of one protein C_{0} |
(0.075–0.2) nm^{−1}^{73,74} |

Area of one protein a_{p} |
(16–70) nm^{2}^{17,19,74} |

Bulk concentration of proteins c_{p} |
(0.1–50) μM^{17,59} |

Dissociation constant K_{D} |
(0.1–5) μM^{59} |

Diffusion coefficient D | (0.01–1) μm^{2} s^{−1}^{75,76} |

Bending rigidity B | (20–40)k_{B}T^{43,74} |

Surface tension λ | (0.003–0.3) × 10^{−3} N m^{−1}^{74,77} |

To understand which process sets the time scale of the membrane dynamics, we perform a scaling analysis. The rate of change of the bending energy can be dissipated by membrane viscosity, i.e., , where E_{b} = B(H − C_{0})^{2} is the bending energy, η_{m} is the membrane viscosity and u is the membrane velocity. The bending energy scales as E_{b} ∼ B/L^{2}, u ∼ L/τ_{v}, where τ_{v} is the viscous time scale and ∇ ∼ 1/L, giving B/τ_{v} ∼ η_{m}L^{2}/τ_{v}^{2}. We can then write τ_{v} ∼ η_{m}L^{2}/B. On the other hand, the diffusive time scale is given by τ_{D} ∼ L^{2}/D, where the diffusion coefficient D is related to the membrane viscosity η_{m} through the Saffman–Delbruck theory,^{78} where . Here, η_{c} is the viscosity of the cytosol and r_{p} ∼ 5 nm is the typical radius of one protein. With the typical values of the membrane and cytosol viscosity in Table 1, the diffusion coefficient . The ratio between these two time scales becomes . With B ∼ 20k_{B}T, we obtain that , that is, the diffusive time scale can be more than one order of magnitude larger than the viscous time scale, supporting our assumption that the mechanical relaxation of the membrane is fast compared to its diffusive transport of proteins.

The equation governing the time evolution of , _{diff} = _{source} in non-dimensional form is written as (see the ESI† for details):

(15) |

The non-dimensional number K_{1} = c_{p}k_{on}L^{2}/D is the ratio between the diffusive time scale and the kinetic recruitment time scale and K_{2} = k_{off}L^{2}/D is the ratio between the diffusive time scale and the protein turnover time scale. is the ratio of the bending energy and the thermal energy. In addition, we set the surface tension λ to be zero, but the influence of λ > 0 is further discussed in the ESI.† The ratio between K_{1} and K_{2} can be written in terms of the dissociation constant K_{D} as . Since we assume the protein density is small as compared to the saturation density, K_{1} and K_{2} must be chosen in such a way that the equilibrium density of proteins in the membrane is small. To reduce the number of parameters influencing the dynamics we set K_{1}/K_{2} = 1/5 in all the numerical simulations, which for _{t} = ′ = 0 = K_{1} − K_{2} gives = 1/5 < 1, consistent with a system where the recruitment is slower that the turnover of proteins. We have chosen B = 20k_{B}T, C_{0} = 0.1 nm^{−1}, and a_{p} = 27 nm^{2}, which gives C_{0} = 5 and Λ = 0.22 in eqn (15) and point out that eqn (8) and (10) are the only shape equations that have a non-dimensional parameter in them, which is C_{0}. The rest of the shape equations, eqn (3)–(5) and (9) are parameter-free. The non-dimensional numbers K_{1} and H_{0} form the basis of a parameter space that will allow us to determine the dependence of the membrane shape with respect to the coupling between diffusion and kinetics.

As we span the phase space of H_{0} ∈ [0.0015–0.15] and K_{1} ∈ [0.2–9] we observe formation of thin membrane necks with respect to the rotational axis r = 0. Since the mathematical model is no longer valid if the neck width is comparable in size with the membrane thickness, we consider the numerical results up to the point when the neck width is equal to the membrane thickness h, that in non-dimensional form has the value h = 0.1 and corresponds to the membrane thickness h reported on Table 1. We define the scission time as t_{cut}. The range of H_{0} corresponds to a radius of curvature of about 300 nm to 30 μm, covering the typical size range of cells (≈6 μm), membrane-bound vesicles (≈500 nm) and giant unilaminar vesicles (up to 200 μm).^{22,79}

Fig. 3 Characteristic membrane shapes at four different snapshots in time when the dimensionless rate coefficient is K_{1} = 4.5 and the threshold for protein recruitment is H_{0} = 0.15. Similarly to the intermediate shapes shown in Fig. 2a–c, we observe that here the membrane shape exhibits a pit-shape (a), U-shape (b) and Ω-shape (c), but at t = t_{cut} a single bud with a constricted neck is formed (d) instead of a pearl, as in Fig. 2d. The color bar represents the protein density (s,t). In (c) and (d) we also observe and almost constant protein density on the vesicle, but it rapidly decays outside of the neck. The scale bar is the dimensionless unit length of the system, equivalent to L = 50 nm. |

To see the details of the protein distribution on the membrane in Fig. 2 and 3, we plot in Fig. 4 the protein density as a function of the membrane area A. In order to clearly illustrate the influence of H_{0} on the protein distribution over the membrane we extract at the same snapshots in time as in Fig. 2 and 3 using the time for membrane scission t_{cut} as a point of reference: t = 0.2t_{cut}, 0.5t_{cut}, 0.8t_{cut} and t_{cut} where [H_{0} = 0.0015, t_{cut} = 0.1] and [H_{0} = 0.15, t_{cut} = 0.38]. A general feature is that the protein density is distributed over a larger area on the membrane when H_{0} is small and it has a smaller gradient as we move from the axis of symmetry to the undeformed membrane. In contrast, when H_{0} is larger the proteins are limited to a much smaller region of the membrane with a steep decay in . By inspecting Fig. 4a–d we can notice the growth rate of the area covered by proteins is much faster when H_{0} is small. To see this, we can roughly estimate the rate of change of covered area ΔA in the time interval Δt = (0.5 − 0.2)t_{cut}, which for H_{0} = 0.0015 is ΔA/Δt ∼ 600 contrasting the same calculation ΔA/Δt ∼ 50 when H_{0} = 0.15. During the last stages of the membrane deformation (Fig. 4b–d) the area covered by proteins increases much slower once an Ω-shape is formed (see Fig. 2b–d and 3b–d) and at the last stage (Fig. 4c and d) this area barely increases and just an increment on the protein density is observed on the membrane for both values of H_{0}. Thus, the geometry of the membrane may also play a role in the dynamic growth process. H_{0} appears to be a critical parameter, because it determines not only the overall size of the budding structures, but also determines in part how proteins are distributed on the membrane. Fig. 4d also shows that as a single vesicle forms (H_{0} = 0.15) the recruited proteins leave the membrane and stalls the dynamics, contrary to when proteins are almost insensitive to the mean curvature and covers a much larger membrane area (H_{0} = 0.0015).

To further characterize qualitatively the membrane dynamics we extract the height of the membrane, z_{max}, along the symmetry axis r = 0 for K_{1} = 4.5 when H_{0} = 0.0015 (Fig. 5a) and H_{0} = 0.15 (Fig. 5b). Initially, we can observe that z_{max} increases as the membrane bud grows in size, but as the membrane starts to form an Ω-shape its height starts to decrease as the neck constricts. The features of z_{max} also allows us to identify the pit-, U- and Ω-shape of the membrane already shown in Fig. 2 and 3. When the membrane forms a pearl-like shape, the bud growth and neck constriction happens several times. The insets in Fig. 5a show the shapes corresponding to the first maximum and the first minimum of z_{max} as well as the final shape at t = t_{cut}. In the time interval between the first maximum and first minimum in z_{max}(t), the membrane shows a gradual transition between a pit-, U- and Ω-shape. As the neck size in the Ω-shape corresponding to the first minimum of z_{max} in Fig. 5a is larger than the typical width h of the membrane bilayer, its height starts to increase again forming a vesicle at the top of the newly formed Ω-shape as we march forward in time. The oscillatory behaviour of z_{max} is not observed when H_{0} = 0.15 (Fig. 5b) but z_{max} has a similar growth and decay when the Ω-shaped membrane is formed.

Next, we turn to map out the membrane shapes predicted by the mathematical model at the scission time, i.e. t = t_{cut}, by systematically varying K_{1} ∈ [0.2–9] and H_{0} ∈ [0.0015–0.15], see Fig. 6. As we go through the parameter space we see how the values of the non-dimensional numbers K_{1} and H_{0} determine if a vesicle buds directly from the membrane or on a deformed foundation as a pit, U or Ω-shape. The phase space in membrane shapes also suggests that we can distinguish membrane deformations that are likely to form only a single vesicle (H_{0} = 0.15) and those that appear to continuously form vesicles by budding from a pit-shape (H_{0} = 0.015, K_{1} = 1.4, 4.5, 9 and H_{0} = 0.0015, K_{1} = 9), U-shape (H_{0} = 0.0015, K_{1} = 4.5) and Ω-shape (H_{0} = 0.0015, K_{1} = 1.4). Qualitatively we can understand the effect of H_{0} by associating this parameter with the membrane region where recruitment of protein occurs: The inverse of H_{0} sets a length scale that becomes large for small H_{0}, then proteins are recruited into a larger portion of the membrane, leading to larger budding structures such as pearls (see Fig. 6), while for larger H_{0} this length scale will become smaller and in this case smaller budding structures would be expected.

The exploration of the phase space spanned by the parameters K_{1} and H_{0} also allows us to determine how they affect the dynamics of the membrane deformation. One quantity that helps to illustrate the time scale of the budding process is t_{cut}. The scission time is measured as the neck size reaches h = 0.1 in the radial direction, see Fig. 7a. In Fig. 7b we present the dependence of t_{cut} with respect to K_{1} in logarithmic axis. Interestingly, we find a universal behaviour: t_{cut} ∼ K_{1}^{−2/3} despite that we vary K_{1} over two orders of magnitude and changing H_{0} only affects the pre-factor of the scaling relation and not the power law. The universal behaviour suggests that the same mechanisms are present across different simulations, where there is a complex interplay between the membrane geometry, diffusion and the area limited for protein recruitment.

To rationalise the power-law dependence we turn to look at the different mechanisms at play. First, we notice that t_{cut} effectively measures the time required for the proteins to cover an area that scales with the typical length of the bud ∼L^{2}. We notice that the growth of this area in time must involve diffusion as it helps increase the area on the membrane in which H > H_{0}. Such a diffusive motion scales by a balance between the two terms on the left hand side of the evolution equation for , eqn (15), which indicates that , or equivalently t_{cut} ∼ L^{2}. On the other hand, within the region where H > H_{0}, the increase of the protein density is kinetically limited and the characteristic length scale is given by L ∼ (C_{0})^{−1}. Since we have for the region with H > H_{0} that , or, ∼ t_{cut}K_{1} and then L ∼ (C_{0}K_{1}t_{cut})^{−1}. By combining these relations between t_{cut} and K_{1}, i.e., t_{cut} ∼ L^{2} ∼ (C_{0}K_{1}t_{cut})^{−2}, we obtain t_{cut} ∼ K_{1}^{−2/3} as predicted by our numerical simulations. Thus, the time scale associated with bud formation, t_{cut}, is a combination of a diffusive front spreading the proteins and the kinetically-limited recruitment process responsible for the local increase of the protein density on the membrane.

At the defined scission time we are now in place to measure the size, r_{bud}, of the vesicle that forms. Fig. 8a reveals that also r_{bud} is a function of K_{1} and H_{0}. We find that when H_{0} is small, the bud radius is sensitive to the parameter K_{1}, but as H_{0} increases, the bud radius becomes insensitive to K_{1} where the formed vesicles have nearly the same size. To understand what sets r_{bud} we turn to the mechanism that drives the dynamics, i.e., the spontaneous curvature induced by the protein density on the membrane. A length scale that appears in our system is 1/(C_{0}_{av}), with is the mean protein density on the bud, i.e. in the region above the membrane neck, and A_{bud} is the area of the membrane comprised between s = 0 and the membrane neck. The vesicle size is now predicted to scale as r_{bud} ∼ 1/(C_{0}_{av}). To test our scaling prediction we scale the bud radius r_{bud} with C_{0}_{av}, which collapses the data onto a single line. To further illustrate the self-similar dynamics in the budding process, we rescale all lengths with the predicted bud size ∼1/(C_{0}_{av}) at the time t_{cut} and shift the profiles so they all start at the same z_{max} at r = 0, see Fig. 8c–e. The upper part of the vesicle all follow the same spherical cap as we would expect from Fig. 8b, but interestingly the profiles map onto a universal shape also in the neck region (inner region) although the far field (outer region) is very different as it takes an Ω, pearl and flat shape. We zoom into the shedding vesicle and plot all the obtained membrane profiles together, which collapses onto a universal shape, shown in Fig. 8f. It is interesting to place this in the context of models for neck closure, where it was recently shown that there are optimal angles formed by the membrane depending on the outer membrane shape with a shape of a dome or a cone.^{80} In the model developed here neck constriction is achieved despite that there is no assembly of specialised scission proteins, but still reveal an optimal angle shared among all shapes Fig. 8f for all the rescaled data K_{1} ∈ [1.35–9], H_{0} ∈ [0.0015–0.15].

We derive a scaling law t_{cut} ∼ K_{1}^{−2/3} based on considering the interplay between the time scale associated with the diffusive spreading of the area allowing protein recruitment and the kinetically limited recruitment process associated with the increase of the local protein density in this area of the membrane. We extract the predicted vesicle size r_{bud} that is a function of both K_{1} and H_{0}, but asymptotes towards a constant vesicle size for K_{1} > 9 where it becomes insensitive to H_{0}. We propose a scaling law for the vesicle size r_{bud} ∼ 1/(C_{0}_{av}) based on the spontaneous curvature induced by the recruited proteins (C_{0}_{av}) where _{av} is the mean protein density in the vesicle. By rescaling the numerical prediction for r_{bud} with this scaling law collapses the data onto a single curve, further highlighting the self-similar budding dynamics.

The membrane shapes predicted by our minimal model can be found in a wide range of biological processes as well as induced by polymers and nanoparticle on lipid vesicles.^{18,81} The mathematical model couples the energy of the membrane to the diffuso-kinetics of the recruited proteins, providing a minimal description of the dynamics. Since the kinetic models describing protein recruitment are phenomenological, as details about the precise binding mechanisms of proteins are, to a large extent, missing in the field, we hope future work can closer couple these and incorporate the statistical mechanics properties as well as viscous flow effects in the recruitment dynamics of curvature sensing proteins. The model proposed here may form a basis for further characterizing how additional biophysical effects e.g., line tension, non-homogeneous bending rigidity and diffusion coefficient and direct protein–protein interactions influence the membrane dynamics.

- L.-G. Wu, E. Hamid, W. Shin and H.-C. Chiang, Annu. Rev. Physiol., 2014, 76, 301–331 CrossRef CAS.
- W. Weissenhorn, E. Poudevigne, G. Effantin and P. Bassereau, Curr. Opin. Virol., 2013, 3, 159–167 CrossRef CAS.
- J. H. Hurley and P. I. Hanson, Nat. Rev. Mol. Cell Biol., 2010, 11, 556–566 CrossRef CAS.
- A. F. Loftus, V. L. Hsieh and R. Parthasarathy, Biochem. Biophys. Res. Commun., 2012, 426, 585–589 CrossRef CAS.
- J. A. Dix and A. Verkman, Annu. Rev. Biophys., 2008, 37, 247–263 CrossRef CAS.
- A. Kenworthy, B. Nichols, C. Remmert, G. Hendrix, M. Kumar, J. Zimmerberg and J. Lippincott-Schwartz, J. Cell Biol., 2004, 165, 735–746 CrossRef CAS.
- N. Jouvenet, M. Zhadina, P. D. Bieniasz and S. M. Simon, Nat. Cell Biol., 2011, 13, 394–401 CrossRef CAS.
- E. M. Wenzel, S. W. Schultz, K. O. Schink, N. M. Pedersen, V. Nahse, A. Carlson, A. Brech, H. Stenmark and C. Raiborg, Nat. Commun., 2018, 9, 2932 CrossRef.
- T. Itoh, K. S. Erdmann, A. Roux, B. Habermann, H. Werner and P. de Camilli, Dev. Cell, 2005, 9, 791–804 CrossRef CAS.
- S. Boulant, C. Kural, J.-C. Zeeh, F. Ubelmann and T. Kirchhausen, Nat. Cell Biol., 2011, 13, 1124–1131 CrossRef CAS.
- L. Lanzetti, Curr. Opin. Cell Biol., 2007, 19, 453–458 CrossRef CAS.
- S. Liese, E. M. Wenzel, R. V. Rojas Molina, S. W. Schultz, H. Stenmark, C. Raiborg and A. Carlson, bioRxiv, 2019 DOI:10.1101/834457.
- M. G. Ford, I. G. Mills, B. J. Peter, Y. Vallis, G. J. Praefcke, P. R. Evans and H. T. McMahon, Nature, 2002, 419, 361–366 CrossRef CAS.
- F. Campelo, H. T. McMahon and M. M. Kozlov, Biophys. J., 2008, 95, 2325–2339 CrossRef CAS.
- P. D. Blood, R. D. Swenson and G. A. Voth, Biophys. J., 2008, 95, 1866–1876 CrossRef CAS.
- J. C. Stachowiak, C. C. Hayden and D. Y. Sasaki, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 7781–7786 CrossRef CAS.
- W. T. Snead, C. C. Hayden, A. K. Gadok, C. Zhao, E. M. Lafer, P. Rangamani and J. C. Stachowiak, Proc. Natl. Acad. Sci. U. S. A., 2017, 114, E3258–E3267 CrossRef CAS.
- I. Tsafrir, D. Sagi, T. Arzi, M. Guedeau-Boudeville, V. Frette, D. Kandel and J. Stavans, Phys. Rev. Lett., 2001, 86, 1138–1141 CrossRef CAS.
- J. C. Stachowiak, E. M. Schmid, C. J. Ryan, H. S. Ann, D. Y. Sasaki, M. B. Sherman, P. L. Geissler, D. A. Fletcher and C. C. Hayden, Nat. Cell Biol., 2012, 14, 944 CrossRef CAS.
- R. Dimova, Adv. Colloid Interface Sci., 2014, 208, 225–234 CrossRef CAS.
- I. Tsafrir, Y. Caspi, M.-A. Guedeau-Boudeville, T. Arzi and J. Stavans, Phys. Rev. Lett., 2003, 91, 138102 CrossRef.
- O. Avinoam, M. Schorb, C. J. Beese, J. A. G. Briggs and M. Kaksonen, Science, 2015, 348, 1369–1372 CrossRef CAS.
- H. McMahon and J. Gallop, Nature, 2005, 438, 590–596 CrossRef CAS.
- M. Kaksonen, C. Toret and D. Drubin, Cell, 2005, 123, 305–320 CrossRef CAS.
- M. Rosendale and D. Perrais, Int. J. Biochem. Cell Biol., 2017, 93, 41–45 CrossRef CAS.
- A. Agrawal and D. J. Steigmann, Z. Angew. Math. Phys., 2011, 62, 549–563 CrossRef CAS.
- J. Derganc, B. Antonny and A. Copic, Trends Biochem. Sci., 2013, 38, 576–584 CrossRef CAS.
- G. Guigas and M. Weiss, Biochim. Biophys. Acta, Biomembr., 2016, 1858, 2441–2450 CrossRef CAS.
- J. Derganc and A. Copic, Biochim. Biophys. Acta, Biomembr., 2016, 1858, 1152–1159 CrossRef CAS.
- W. T. Gozdz, J. Chem. Phys., 2011, 134, 024110 CrossRef CAS.
- R. Lipowsky, Faraday Discuss., 2013, 161, 305 RSC.
- P. Rangamani, K. K. Mandadap and G. Oster, Biophys. J., 2014, 107, 751–762 CrossRef CAS.
- C. Tozzi, N. Walani and M. Arroyo, New J. Phys., 2019, 21, 093004 CrossRef CAS.
- A. Mahapatra, D. Saintillan and P. Rangamani, bioRxiv, 2020 DOI:10.1101/2020.01.14.906917.
- J. S. Sohn, Y.-H. Tseng, S. Li, A. Voigt and J. S. Lowengrub, J. Comput. Phys., 2010, 229, 119–144 CrossRef CAS.
- J. Lowengrub, J. Allard and S. Aland, J. Comput. Phys., 2016, 309, 112–128 CrossRef CAS.
- K. E. Teigen, X. Li, J. Lowengrub, F. Wang and A. Voigt, Commun. Math. Sci., 2009, 7, 1009–1037 CrossRef.
- W. Helfrich, Z. Naturforsch., C: J. Biosci., 1973, 28, 693–703 CAS.
- P. B. Canham, J. Theor. Biol., 1970, 26, 61–81 CrossRef CAS.
- H. Deuling and W. Helfrich, Biophys. J., 1976, 16, 861–868 CrossRef CAS.
- H. Alimohamadi and P. Rangamani, Biomolecules, 2018, 8, 120 CrossRef.
- D. Schley, R. J. Whittaker and B. W. Neuman, J. R. Soc., Interface, 2013, 10, 20130403 CrossRef.
- S. Dmitrieff and F. Nedelec, PLoS Comput. Biol., 2015, 11, 1–15 CrossRef.
- L. Foret, Eur. Phys. J. E: Soft Matter Biol. Phys., 2014, 37, 42 CrossRef.
- H. Alimohamadi, B. Ovryn and P. Rangamani, Sci. Rep., 2020, 10, 1–15 CrossRef.
- M. Arroyo, N. Walani, A. Torres-Sánchez and D. Kaurin, Onsager's Variational Principle in Soft Matter: Introduction and Application to the Dynamics of Adsorption of Proteins onto Fluid Membranes, in The Role of Mechanics in the Study of Lipid Bilayers. CISM International Centre for Mechanical Sciences (Courses and Lectures), ed. D. Steigmann, Springer, Cham, 2018 Search PubMed.
- A. Torres-Sanchez, D. Millan and M. Arroyo, J. Fluid Mech., 2019, 872, 218–271 CrossRef CAS.
- N. S. Gov, Philos. Trans. R. Soc., B, 2018, 373, 20170115 CrossRef.
- A. Veksler and N. S. Gov, Biophys. J., 2007, 93, 3798–3810 CrossRef CAS.
- R. Lipowsky, Faraday Discuss., 2013, 161, 305–331 RSC.
- R. Lipowsky, Adv. Colloid Interface Sci., 2014, 208, 14–24 CrossRef CAS.
- C.-H. Chang and E. I. Franses, Colloids Surf., A, 1995, 100, 1–45 CrossRef CAS.
- K. Foo and B. Hameed, Chem. Eng. J., 2010, 156, 2–10 CrossRef CAS.
- S. Leibler, J. Phys., 1986, 47, 507–516 CrossRef CAS.
- H. Jian-Guo and O.-Y. Zhong-Can, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 1993, 47, 461–467 CrossRef.
- M. Deserno, Notes on differential geometry, 2004 Search PubMed.
- U. Seifert, K. Berndl and R. Lipowsky, Phys. Rev. A: At., Mol., Opt. Phys., 1991, 44, 1182–1202 CrossRef CAS.
- S. P. Preston, O. E. Jensen and G. Richardson, Q. J. Mech. Appl. Math., 2008, 61, 1–24 CrossRef.
- E. M. Smith, J. Hennen, Y. Chen and J. D. Mueller, Biophys. J., 2015, 108, 2648–2657 CrossRef CAS.
- P. M. Chaikin and T. C. Lubensky, Principles of condensed matter physics, Cambridge University Press, 1995 Search PubMed.
- G. B. Arfken, H. J. Weber and F. E. Harris, Mathematical methods for physicists, Academic Press, 7th edn, 2013 Search PubMed.
- T. Masuda, H. Hirose, K. Baba, A. Walrant, S. Sagan, N. Inagaki, T. Fujimoto and S. Futaki, Bioconjugate Chem., 2020, 31, 1611–1615 CrossRef CAS.
- J. B. Larsen, M. B. Jensen, V. K. Bhatia, S. L. Pedersen, T. Bjornholm, L. Iversen, M. J. Uline, I. Szleifer, K. J. Jensen, N. S. Hatzakis and D. Stamou, Nat. Chem. Biol., 2015, 11, 192–194 CrossRef CAS.
- W. Zhao, L. Hanson, H.-Y. Lou, M. Akamatsu, P. D. Chowdary, F. Santoro, J. R. Marks, A. Grassart, D. G. Drubin, Y. Cui and B. Cui, Nat. Nanotechnol., 2017, 12, 750–756 CrossRef CAS.
- C. Florentsen, A. Kamp-Sonne, G. Moreno-Pescador, W. Pezeshkian, A. A. H. Zanjani, H. Khandelia, J. Nylandsted and P. M. Bendix, Soft Matter, 2020 10.1039/d0sm00241k.
- T. V. S. Krishnan, S. L. Das and P. B. S. Kumar, Soft Matter, 2019, 15, 2071–2080 RSC.
- J. Liu, Y. Sun, D. G. Drubin and G. F. Oster, PLoS Biol., 2009, 7, 1–16 Search PubMed.
- P. A. Janmey and P. K. J. Kinnunen, Trends Cell Biol., 2006, 16, 538–546 CrossRef CAS.
- P. Cicuta, S. L. Keller and S. L. Veatch, J. Phys. Chem. B, 2007, 111, 3328–3331 CrossRef CAS.
- P. Sens, Phys. Rev. Lett., 2004, 93, 108103 CrossRef.
- A. Carlson and L. Mahadevan, PLoS Comput. Biol., 2015, 11, 1–16 CrossRef.
- D. Wirtz, Annu. Rev. Biophys., 2009, 38, 301–326 CrossRef CAS.
- J. Liu, R. Tourdot, V. Ramanan, N. J. Agrawal and R. Radhakrishanan, Mol. Phys., 2012, 110, 1127–1137 CrossRef CAS.
- Z. Shi and T. Baumgart, Nat. Commun., 2015, 6, 5974 CrossRef CAS.
- D. Thalmeier, J. Halatek and E. Frey, Proc. Natl. Acad. Sci. U. S. A., 2016, 113, 548–553 CrossRef CAS.
- J. Liu, Y. Sun, G. F. Oster and D. G. Drubin, Curr. Opin. Cell Biol., 2010, 22, 36–43 CrossRef CAS.
- M. M. Kozlov and L. V. Chernomordik, Curr. Opin. Struct. Biol., 2015, 33, 61–67 CrossRef CAS.
- P. G. Saffman and M. Delbrück, Proc. Natl. Acad. Sci. U. S. A., 1975, 72, 3111–3113 CrossRef CAS.
- P. Bassereau, R. Jin, T. Baumgart, M. Deserno, R. Dimova, V. A. Frolov, P. V. Bashkirov, H. Grubmueller, R. Jahn, H. J. Risselada, L. Johannes, M. M. Kozlov, R. Lipowsky, T. J. Pucadyil, W. F. Zeno, J. C. Stachowiak, D. Stamou, A. Breuer, L. Lauritsen, C. Simon, C. Sykes, G. A. Voth and T. R. Weikl, J. Phys. D: Appl. Phys., 2018, 51, 343001 CrossRef.
- J. Agudo-Canalejo and R. Lipowsky, PLoS Comput. Biol., 2018, 14, 1–21 CrossRef.
- Y. Yu and S. Granick, J. Am. Chem. Soc., 2009, 131, 14158–14159 CrossRef CAS.

## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0sm01028f |

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