Curvature-sensing and generation by membrane proteins: a review

Abstract

Membrane proteins are crucial in regulating biomembrane shapes and controlling the dynamic changes in membrane morphology during essential cellular processes. These proteins can localize to regions with their preferred curvatures (curvature sensing) and induce localized membrane curvature. Thus, this review describes the recent theoretical development in membrane remodeling performed by membrane proteins. The mean-field theories of protein binding and the resulting membrane deformations are reviewed. The effects of hydrophobic insertions on the area-difference elasticity energy and that of intrinsically disordered protein domains on the membrane bending energy are discussed. For the crescent-shaped proteins, such as Bin/Amphiphysin/Rvs superfamily proteins, anisotropic protein bending energy and orientation-dependent excluded volume significantly contribute to curvature sensing and generation. Moreover, simulation studies of membrane deformations caused by protein binding are reviewed, including domain formation, budding, and tubulation.

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Hiroshi Noguchi

Hiroshi Noguchi received his PhD degree at Nagoya University in 2000. He worked at the Institute for Molecular Science for three years and at Forschungszentrum Juelich for five years. He has been an associate professor at the University of Tokyo since 2008. The main research focus is understanding dynamics of biomembranes and complex fluids using theory and coarse-grained simulations.

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1 Introduction

Cell membranes and organelles exhibit a variety of shapes. Various types of proteins are known to control dynamical changes in membrane morphology during essential cellular processes, such as endocytosis, exocytosis, vesicle transport, mitosis, and cell locomotion.^1–7 During in vitro experiments, protein binding has been observed to induce membrane budding and tubulation. Additionally, membrane proteins can localize to membrane regions of specific curvature. These two phenomena are referred to as curvature generation and curvature sensing, respectively. This review focuses on the theoretical studies of protein behaviors with emphasis on thermal equilibrium and relaxation to the equilibrium. Non-equilibrium membrane dynamics, such as non-thermal fluctuations^8–11 and wave propagations,^12–18 are covered in our recent review.¹⁹ Moreover, because membrane simulation models and methods have previously been reviewed in ref. 20–26, this review primarily describes the mean-field theory and presents relevant simulation results without delving into detailed simulation methodologies.

Section 2 provides an overview of the bending energy of lipid membranes and their morphology in the absence of proteins. Section 3 discusses curvature-sensing. Certain proteins exhibit laterally isotropic shapes in a membrane and bend the membrane isotropically. Section 3.1 presents the theoretical aspects of isotropic proteins, and Section 3.2 explores how intrinsically disordered protein (IDP) domains influence membrane bending properties. Section 3.3 addresses the behavior of anisotropic proteins. Crescent-shaped proteins, such as Bin/Amphiphysin/Rvs (BAR) superfamily proteins, induce membrane bending along their major protein axes. Section 3.4 examines protein binding to tethered vesicles and presents the estimation of protein bending properties. Section 4 focuses on curvature generation, with Sections 4.1 and 4.2 reviewing membrane deformations induced by the isotropic and anisotropic proteins, respectively. Section 4.3 discusses the membrane deformation by the adhesion of colloidal nanoparticles. Finally, Section 5 provides a summary and outlook.

2 Lipid membranes

In a fluid phase, lipid membranes are laterally isotropic, and their bending energy can be expressed using the second-order expansion of the membrane curvature, known as the Canham–Helfrich model.^27,28where A represents the membrane area. The membrane mean and Gaussian curvatures are defined as H = (C₁ + C₂)/2 and K = C₁C₂, respectively, where C₁ and C₂ represent the principal curvatures (see Fig. 1). The coefficients κ_d and [small kappa, Greek, macron]_d denote the bending rigidity and saddle-splay modulus (also referred to as the Gaussian curvature modulus), respectively. The parameter C_mb denotes the spontaneous curvature. Note that the spontaneous curvature is often expressed as H_mb = C_mb/2, which is particularly useful in the analysis of spherical membranes, while C_mb is useful for cylindrical membranes. For lipid bilayers with symmetric leaflets, the membrane has zero spontaneous curvature (C_mb = 0). The last term in eqn (1) can be neglected when considering the shape transformation of vesicles with a fixed topology, owing to the Gauss–Bonnet theorem, where g represents the genus of the vesicle. Lipid membranes typically exhibit a bending rigidity in the range of κ_d = 10–100 k_BT^29–32 and [small kappa, Greek, macron]_d/κ_d ≃ −1³³, where k_BT is the thermal energy. In this review, we use κ_d = −[small kappa, Greek, macron]_d = 20 k_BT and C_mb = 0, unless otherwise specified.

Fig. 1 Schematic of lipid membranes. A membrane locally bends with two principal curvatures C₁ and C₂. A spherical membrane has C₁ = C₂ = 1/R_sp (H = 1/R_sp and K = 1/R_sp²), while a cylindrical membrane has C₁ = 1/R_cy and C₂ = 0 (H = 1/2R_cy and K = 0).

In lipid membranes, the traverse movement of phospholipids between the two leaflets, known as flip–flop, occurs at an extremely slow rate, with half-lives ranging from hours to days.³⁴ In contrast, amphiphilic molecules with small hydrophilic head groups, such as cholesterols, exhibit significantly faster flip–flop dynamics, occurring within seconds to minutes.^35–37 In living cells, proteins facilitate flip–flop. Flippase and floppase proteins actively transport specific lipids from the outer to the inner leaflets (flip) or in the opposite direction (flop), respectively, through ATP hydrolysis, leading to an asymmetric lipid distribution. Conversely, scramblases mediate the bidirectional translocation of lipids, allowing the bilayer to relax toward a thermal-equilibrium lipid distribution.³⁵ As a result, the number of lipids in each leaflet remains constant over typical experimental timescales, although it can relax with the addition of cholesterols,^38,39 ultra-long-chain fatty acids,^40,41 and scramblases. Consequently, the area difference of the two leaflets in a liposome may differ from the lipid-preferred area difference ΔA₀ = (N_out − N_in)a_lip, where N_out and N_in represent the numbers of lipids in the outer and inner leaflets, respectively, a_lip is the area per lipid, and h ≃ 2 nm denotes the distance between the centers of the two leaflets. In the area difference elasticity (ADE) model,^42–44 the energy associated with the mismatch ΔA − ΔA₀ is accounted for by a harmonic potential:

= 8πk_r(Δa − Δa₀)², (4)

where the coefficient k_r = πk_ade. In eqn (3) and (4), the area differences are normalized as m = ΔA/2hR_A and Δa = ΔA/8πhR_A, where the lengths are normalized using the vesicle surface area as R_A = (A/4π)^1/2. These two formulations were used in ref. 42 and ref. 43 and 44, respectively. For typical lipid membranes, k_ade ≃κ_d was estimated.⁴⁵

Because the critical micelle concentration (CMC) of lipids is extremely low,⁴⁶ the number of lipid molecules within a vesicle remains essentially constant over typical experimental timescales. Additionally, the internal volume is maintained nearly constant due to osmotic pressure, since water molecules can slowly permeate the lipid bilayer, whereas the penetration of ions or macromolecules is negligible. Under the constraints of a constant volume V and constant surface area A at C_mb = 0, the global energy minimum of F_cv0 corresponds to different vesicle shapes depending on the reduced volume V* = 3V/(4πR_A³). In the mechanical (force) viewpoint, the stress caused by the bending energy is balanced by the surface tension (γ) and the osmotic pressure difference between the inner and outer solutions. For vesicles with genus g = 0, stomatocyte, discocyte, and prolate shapes achieve global energy minima within the ranges 0 < V* ≲ 0.59, 0.59 ≲ V* ≲ 0.65, and 0.65 ≲ V* < 1, respectively.^42,47,48 These three shapes can coexist as (meta-)stable states at V* ≃ 0.6,⁴⁹ and the prolate shape can persist as a meta-stable state even at V* ≲ 0.6, as illustrated in Fig. 2.^50,51 Note that red blood cells have a discocyte shape with V* ≃ 0.6 in the physiological condition, and their membranes have shear elasticity due to the cytoskeletons underneath the membranes.⁵² When the ADE energy is included, additionally, branched tubular vesicles and budding (where spherical buds form on the outside of a spherical vesicle)^42,53 emerge alongside the stomatocyte, discocyte, and prolate shapes. Notably, experimental observations have been well reproduced by this theoretical model.⁴⁵ Furthermore, rapid changes in ΔA₀ induced by chemical reactions and other factors can lead to the protrusion of bilayer sheets, reducing the area difference.^54,55

Fig. 2 Stable and meta-stable shapes of vesicles in the Canham–Helfrich model (eqn (1)) with C_mb = 0.^49,50 (a) Snapshots obtained by dynamically-triangulated MC simulations. Stomatocyte at V* = 0.5, discocyte at V* = 0.6, and prolate at V* = 0.5. (b) Area difference Δa of (meta-) stable shapes. Adapted from ref. 50 with permission from the Royal Society of Chemistry (2015).

3 Curvature sensing

Peripheral and transmembrane proteins tend to accumulate in membrane regions that match their preferred curvatures. The surface densities of peripheral proteins are governed by the balance between the protein binding and unbinding processes in thermal equilibrium. In contrast, in typical in vitro experiments, the total number of transmembrane proteins within a vesicle remains fixed. These scenarios correspond to grand canonical and canonical ensembles in the membrane, respectively. While the choice between these two conditions does not affect average properties, such as surface protein density and alignment, it influences kinetics and fluctuations (the second derivatives of free energy).

3.1 Theory of isotropic proteins

First, we discuss the curvature-sensing phenomenon of proteins with a laterally isotropic shape. The insertion of a hydrophobic α-helix and the anchoring of IDP domains do not exhibit a preferred bending direction. Moreover, proteins or protein assemblies possessing threefold, fivefold, or higher rotational symmetry exhibit laterally isotropic bending energy, when their asymmetric deformations are negligible.⁵⁶ Several types of ion channels^57,58 and G-protein coupled receptors (GPCRs),^59–62 have rotationally symmetric structures. For instance, the trimer and pentamer of microbial rhodopsins exhibit threefold and fivefold symmetry, respectively.⁶² Furthermore, certain peripheral proteins, such as a clathrin monomer⁶³ and annexin A5 trimer,^64,65 also possess threefold symmetry.

The presence of membrane-bound proteins can alter the membrane bending rigidity and spontaneous curvature relative to a bare (unbound) membrane. The bending energy of a vesicle can be expressed as⁶⁶

where κ_pi, [small kappa, Greek, macron]_pi, and C₀ = 2H₀ denote the bending rigidity, saddle-splay modulus, and spontaneous curvature of the protein-occupied membrane, respectively, and ϕ represents the local protein density (area fraction, i.e., ϕ = 1 indicates complete coverage). This formulation accounts for the bending energy induced by the protein–membrane interactions. Additionally, inter-protein interactions – such as the steric effects arising from the brush region of IDP chains discussed in Section 3.2 – can further influence membrane rigidity and spontaneous curvature. In that case, κ_pi, [small kappa, Greek, macron]_pi, and C₀ become functions of ϕ.⁶⁶

At κ_pi > κ_d, the bending energy can also be expressed as F_cv1 = F_cv0 + F_pi with

where κ_pi = κ_pa + κ_d, [small kappa, Greek, macron]_pi = [small kappa, Greek, macron]_pa + [small kappa, Greek, macron]_d and C₀ = [κ_pa/(κ_pa + κ_d)]C_0a. This formulation is known as the curvature mismatch model, where κ_pa represents the additional bending rigidity by protein binding, while κ_pi accounts for the combined rigidity of the protein and the underlying membrane. The curvature mismatch model with [small kappa, Greek, macron]_pa = 0 was used in ref. 67–69.

The total free energy F of a vesicle consists of the bending energy F_cv1, the inter-protein interaction energy, and the mixing entropy:

where a_p denotes the area occupied by a single protein. The second term in eqn (7) represents the pairwise inter-protein interactions, where b < 0 and b > 0 indicate attractive and repulsive interactions between bound proteins, respectively. The third term in eqn (7) accounts for the mixing entropy of the bound proteins.

The binding equilibrium of peripheral proteins is determined by minimizing J = F − μN, where μ is the binding chemical potential of the protein binding, and is the number of the bound proteins. Consequently, the local protein density ϕ is given by ∂f/∂ϕ = μ/a_p, where . When the inter-protein interactions are negligible (b = 0), ϕ is expressed by a sigmoid function of μ:⁶⁶

This relation reflects the detailed balance between protein binding and unbinding at a local membrane region: η_ub/η_b = exp(w_b) for the kinetic equation dϕ/dt = η_b(1 − ϕ) − η_ubϕ.^70,71 For b ≠ 0, ϕ can be solved iteratively by replacing w_b with w_b + 2bϕa_p/k_BT in eqn (8).⁶⁶

For κ_pi > κ_d (κ_pa > 0), the protein density ϕ exhibits a peak at a finite curvature (referred to as the sensing curvature H_s, see Fig. 3(a)). The maximum value of ϕ increases from 0 to 1 with increasing μ. Notably, the protein binding differ between spherical and cylindrical membranes with the same mean curvature H, when [small kappa, Greek, macron]_pi ≠ [small kappa, Greek, macron]_d. The proteins bind more to spherical membranes compared to cylindrical membranes at ([small kappa, Greek, macron]_pi − [small kappa, Greek, macron]_d)/(κ_pi − κ_d) = −1 (see Fig. 3(a)). The sensing curvature H_s is obtained by solving dϕ/dH = 0 using eqn (8) under the conditions K = H² for spherical membranes and K = 0 for cylindrical membranes:

where κ_dif = κ_pi − κ_d + ([small kappa, Greek, macron]_pi − [small kappa, Greek, macron]_d)/2 and κ_dif = κ_pi − κ_d for the spherical and cylindrical membranes, respectively. As κ_pi increases from κ_d to ∞, H_s decreases from ∞ to C₀/(2 + [small kappa, Greek, macron]_pi/κ_pi) for spherical membranes and to C₀/2 for cylindrical membranes (see Fig. 3(b)). It is important to note that ϕ(H) is mirror symmetric with respect to the sensing curvature for both cylindrical and spherical membranes (see Fig. 3(a)).

Fig. 3 Curvature sensing of isotropic proteins at C₀²a_p = 0.04 (e.g., C₀ = 0.02 nm⁻¹ for a_p = 100 nm²) and [small kappa, Greek, macron]_pi/κ_pi = [small kappa, Greek, macron]_d/κ_d = −1.^66,70 (a) Protein density ϕ as a function of the local mean curvature H at κ_pi/κ_d = 0.5, 1, 2, and 4 for μ = 0. (b) Sensing curvature H_s and the maximum generation curvature H_g as a function of bending rigidity ratio κ_pi/κ_d. H_s and H_g are given by eqn (10) and (30) with ϕ = 1, respectively. The solid and dashed lines represent the data for spherical and cylindrical membranes, respectively.

In contrast, for κ_pi < κ_d, the bound membrane exhibits a lower bending rigidity compared to the bare membrane. This scenario may arise when the bound proteins (or other molecules) remodel the bound membrane. For example, a reduction in membrane thickness can lead to decreased bending rigidity. Interestingly, the proteins hold a negative curvature sensing at κ_pi < κ_d, where ϕ exhibits a minimum instead of a maximum (see the gray lines in Fig. 3(a)).⁷⁰ In other words, the fraction of bare membrane, 1 − ϕ, reaches its maximum at the negative sensing curvature. Owing to the lower rigidity, the bound membranes bend passively to reduce the bending energy of bare membrane regions, sometimes even in the opposite direction to their spontaneous curvatures. Consequently, these proteins cannot induce membrane bending to their spontaneous curvatures. Therefore, a higher bending rigidity (κ_pi > κ_d) is required to bend membranes to a specific curvature.

For κ_pi = κ_d, ϕ follows a monotonic sigmoid function of H without any distinct peaks (see the green line in Fig. 3(a)). In several previous studies,^72–75 the condition κ_pi = κ_d was set as a simplified model, and the following bending energy was used:

This formulation corresponds to the condition of κ_pi = κ_d, [small kappa, Greek, macron]_pi = [small kappa, Greek, macron]_d, and b = κ_dC₀²/2. The quadratic term (κ_dC₀²/2)ϕ² is often neglected.^76,77 Since this quadratic term is independent of membrane curvature and represents a pairwise inter-protein interaction, its inclusion in the bending energy is not recommended. Similarly, preaveraging both bending rigidity and spontaneous curvature as is not advisable, because it implicitly accounts for pairwise and three-body inter-protein interactions ((2κ₁C₀H + κ_dC₀²/2)ϕ² and (κ₁C₀²/2)ϕ³, respectively).^70,78 Although the previous studies^69,74,79 have compared the two models given by eqn (6) and (11) as distinct approaches, they are, in fact, the subsets of eqn (6) for κ_pi ≠ κ_d and κ_pi = κ_d, respectively.

The chemical potential μ can be modulated by adjusting the bulk protein concentration ρ. For a dilute solution, it is expressed as μ(ρ) = μ(1) + k_BT ln(ρ). In experiments, the ratio of surface protein densities at different curvatures has often been used, making the estimation of μ unnecessary. For a large spherical vesicle with R_AC₀ ≫ 1, the membrane can be approximated as flat (H = K = 0), and the protein density is given by ϕ_flat = 1/{1 + exp[(−μ + a_pκ_piC₀²/2)/k_BT]} for b = 0. Hence, for the protein density ϕ_cy in a cylindrical membrane with radius R_cy, eqn (8) can be rewritten as⁸⁰

In the low-density limit (ϕ_flat ≪ 1 and ϕ_cy ≪ 1), the density ratio is simplified to an exponential function as^67,81

for the bending-energy formulations given in eqn (5) and (6), respectively. In this limit, the ratio ϕ_cy/ϕ_flat is independent of ϕ_flat.

3.2 Intrinsically disordered protein (IDP) domains

Many curvature-inducing proteins contain IDP domains. Stachowiak and coworkers have investigated the effects of varying the length of IDP domains in BAR and other proteins and have reported that the disordered domains facilitate curvature sensing and that the longer IDP chains promote the formation of small vesicles.^82–85 A disordered domain behaves as a linear polymer chain in a good solvent,⁸⁶ that is, its mean radius of gyration scales as 〈R_g〉 ∼ n^ν_poly, where 〈⋯〉 denotes the ensemble average, n_poly represents the number of Kuhn segments, and ν = 0.6 is the scaling exponent for an excluded volume chain.^87,88 The Kuhn length of IDP chains is approximately 0.8 nm.⁸⁶ The interactions between membrane-anchored polymer chains and membrane have been extensively studied through theory,^89–93 simulations,^94–97 and experiments.^98–102 The formation of spherical buds^98,99,101 and membrane tubes^99,100,102 have been observed experimentally. Polymer anchoring induces a positive spontaneous curvature of the membrane and increases the bending rigidity in a good solvent.

At low polymer densities (referred to as the “mushroom regime”), the polymer chain exists in isolation on the membrane forming a mushroom-like distribution, where the inter-polymer interactions are negligible. In this regime, both the spontaneous curvature and bending rigidity increase linearly with the grafting density ϕ_poly of the polymer chains. Analytically, the relations^89,90

κ_piΔH₀ = k_h0k_BTR_endϕ_poly, (14)

Δκ = k_κk_BTR_end²ϕ_poly, (15)

Δ[small kappa, Greek, macron] = −k̄_κk_BTR_end²ϕ_poly, (16)

are predicted, where ΔH₀ = ΔC₀/2, Δκ, and Δ[small kappa, Greek, macron] represent the differences in the spontaneous curvatures, bending rigidities, and saddle-splay moduli between the polymer-decorated and bare membranes, respectively, and R_end represents the mean end-to-end distance of the polymer chain. These coefficients have been analytically derived using Green's function for ideal chains^89,90 and have also been estimated by Monte Carlo (MC) simulations of single anchored polymer chains:⁹⁴k_h0 = 0.18 and 0.17; k_κ = 0.21 and 0.2; and k̄_κ = 0.17 and 0.15; for ideal and excluded-volume chains, respectively.

At a polymer density sufficiently higher than the overlap density (referred to as “brush regime”), the polymer chains extend perpendicularly from the membrane surface, forming a brush-like structure. In this regime, polymers grafting further enhance both the bending rigidity and spontaneous curvature of the membrane. In the limit of small curvature, the bending rigidity and saddle-splay modulus are given by⁹⁰

where is the polymer density normalized by the maximum coverage. Consequently, brush polymers increases the membrane rigidity in proportion to ϕ^2.5_poly, as show in Fig. 4.

Fig. 4 Schematic graph of the bending modulus modification by IDP chains. The red and blue lines represent the differences in the bending rigidities (Δκ), and saddle-splay moduli (Δ[small kappa, Greek, macron]) between the IDP-decorated and bare membranes, respectively. At sufficiently lower and higher densities than the overlap density (ϕ_overlap), the IDP chains exhibit mushroom and brush shapes, respectively, as drawn in the inset. In the mushroom region, Δκ and Δ[small kappa, Greek, macron] are expressed by eqn (15) and (16), respectively, with the coefficients for a good solvent.⁹⁴ In the brush region, Δκ and Δ[small kappa, Greek, macron] are expressed by eqn (17) and (18), respectively, in the small curvature limit.⁹⁰

In addition, polymer grafting reduces the line tension of membrane edges, thereby stabilizing the microdomains with a size of the polymer-chain length.⁹⁷ Furthermore, in a poor solvent environment, the polymer grafting can induce a negative spontaneous curvature, leading to the formation of a dimple-shaped membrane structure.^96,103

3.3 Theory of anisotropic proteins

Here, we consider the binding of anisotropic proteins to membranes. A prominent example of anisotropic proteins is the BAR superfamily proteins, which features a banana-shaped binding domain known as the BAR domain. This binding domain is a dimer and holds twofold rotational symmetry. The BAR domain binds to the membrane, inducing curvature along the domain axis and generating cylindrical membrane tubes.^{1,3,73,104–109} The BAR domains have lengths from 13 to 27 nm.¹⁰⁴ N-BAR and F-BAR domains have a positive curvature along the domain axis and I-BAR (Inverse-BAR) domains have a negative curvature. Some BAR proteins also have extra binding domains, such as phox homology (PX) and pleckstrin homology (PH) domains, and N-BAR and some extra domains have membrane insertion modules.^105,110,111 These domains and modules can modify the bending energy of bound regions, with maintaining the rotational symmetry.

Not all curvature-inducing proteins exhibit rotational symmetry. For example, dynamin,^112–114 which has an asymmetric structure, forms helical assemblies that constrict membrane neck, leading to membrane fission. Similarly, melittin and amphipathic peptides^115–118 bind to membranes, and their circular assemblies result in membrane pore formation. Recent coarse-grained simulation of a buckled membrane by Gómez-Llobregat and coworkers demonstrated the curvature sensing of three amphipathic peptides.¹¹⁹ They revealed that melittin and the amphipathic peptides LL-37 (PDB: 2k6O) exhibited asymmetric curvature sensing, meaning that their angular distribution relative to the buckled axis is not symmetric.

Several bending-energy models have been proposed to describe the behavior of anisotropic proteins. For the crescent-shaped symmetric proteins, such as BAR proteins, the bending energy can be expressed as^70,120,121

where κ_p and C_p represent the bending rigidity and spontaneous curvature along the major protein axis, respectively, while κ_s and C_s denote those along the minor (side) axis. The membrane curvatures along these major and minor axes are given by

C_{[small script l]m} = C₁ cos²(θ_pc) + C₂ sin²(θ_pc) = H + D cos(2θ_pc), (20)

C_{[small script l]s} = C₁ sin²(θ_pc) + C₂ cos²(θ_pc) = H − D cos(2θ_pc), (21)

respectively, where D = (C₁ − C₂)/2 represents the deviatoric curvatures of the membrane (D² = H² − K), and θ_pc represents the angle between the protein axis and the direction of one of the principal membrane curvatures (typically, the azimuthal direction is selected for cylindrical membranes). The protein bends the membrane with κ_p and C_p along the major axis and with κ_s and C_s along the minor axis. If the side regions of the linear-shaped proteins bind strongly to the membrane, they exhibit a negative side curvature (C_s < 0).^122,123 Conversely, the excluded-volume repulsion between adjacent proteins can generate a positive side curvature.¹²⁴

A protein can comprise binding domains with distinct bending axes (where C_{[small script l]j} denotes the membrane curvature along the axis of the j-th domain) and isotropic bending regions (IDP domains etc.). Consequently, the bending energy of a single protein is generally expressed as⁵⁶

= k₁H² + k₂H + k₃K + k₄D cos(2θ_pc) + k₅HD cos(2θ_pc) + k₆D² cos(4θ_pc) + k₇D sin(2θ_pc) + k₈HD sin(2θ_pc) + k₉D² sin(4θ_pc) + U₀, (23)

in the second-order expansion of membrane curvature. The constant term U₀ can be neglected by incorporating it into the chemical potential, such that μ′ = μ + U₀. Isotropic proteins are characterized by the first three terms with the coefficients k₁ = 2κ_paa_p, k₂ = −2κ_paa_pC_0a, and k₃ = [small kappa, Greek, macron]_paa_p for F_pi (compare eqn (6) and (23)). Proteins with twofold rotational or mirror symmetry can have the first six terms (k₁–k₆), while asymmetric proteins may exhibit all nine terms. The protein major axis can be chosen to be k₇ = 0 in order to reduce the number of coefficients. To express asymmetry, k₈ ≠ 0 or k₉ ≠ 0 is needed at this axis. The protein model in eqn (19) is considered with N_ax = 2 and F_pi = 0, assuming orthogonal axes where C_{[small script l]1} = C_{[small script l]m} and C_{[small script l]2} = C_{[small script l]s}, and the coefficients are mapped accordingly as k₁ = 3(κ_p + κ_s)a_p/4, k₂ = −(κ_pC_p + κ_sC_s)a_p, k₃ = −(κ_p + κ_s)a_p/4, k₄ = −(κ_pC_p − κ_sC_s)a_p, k₅ = (κ_p − κ_s)a_p, and k₆ = (κ_p + κ_s)a_p/4.⁵⁶ Akabori and Santangelo¹²⁵ have added U_asy = k_asy[D sin(2θ_pc) − C_asy]² to eqn (19) in order to include an asymmetric bending effect. Their formulation corresponds to eqn (23) with k₇ = −2k_asyC_asy and k₈ = k₉ = 0, modifying k₁, k₃, and k₆. Kralj-Iglič and coworkers have considered the protein energy with a symmetric shape,^126,127U_p = k_a(H − H₀)²/2 + (k_a + k_b)[D² − 2DD₀ cos(2θ_pc) + D₀²]/4. The second term assumes an energy proportional to a rotational average of (d(C_{[small script l]} − C_m)/dθ)², where C_{[small script l]} is the normal membrane curvature at the angle θ, and C_m = C_m0 + C_m1 cos(2θ) is the angle-dependent spontaneous curvature. In this formulation, k₁ = 3k_a/4 + k_b/4, k₂ = −k_aH₀, k₃ = −(k_a + k_b)/4, k₄ = −(k_a + k_b)D₀/2, and k₅ = k₆ = 0. They have also used the first term of eqn (19) for rod-like proteins (i.e., κ_s = 0).¹²⁷ Fournier combined an anisotropic bending energy with the tilt energy of lipids for transmembrane proteins.¹²⁸

3.3.1 Isolated proteins. First, we consider protein binding in the low-density limit, in which bound proteins are isolated on the membrane and inter-protein interactions are negligible. Hence, the density ϕ of bound proteins is given by . The binding ratio of proteins to a cylindrical membrane tube relative to a flat membrane is expressed as⁵⁶where U^flat_p is the bending energy for the flat membrane, and U^cy_p is that for the cylindrical membrane. This ratio ϕ_cy/ϕ_flat is independent of μ in the low-density limit, as in the isotropic proteins.

Anisotropic proteins can adjust their lateral orientation to reduce their bending energy. Let us consider a crescent symmetric protein (eqn (19) with κ_s = 0) and its variants as simple anisotropic protein models. This crescent protein has the lowest bending energy at θ_pc = 0 (the protein orients in the azimuthal direction) in a wide cylinder (1/R_cyC_p ≤ 1), whereas tilt proteins have the lowest at in a narrow cylinder (1/R_cyC_p > 1). Hence, the protein density exhibits peaks at these preferred orientations (see the red lines in Fig. 5(b) and (c)). The average density ϕ_cy also exhibits a peak at a membrane curvature slightly higher than 1/R_cyC_p = 1 (see Fig. 5(a)). Unlike isotropic proteins, ϕ_cy(R_cy) is not mirror symmetric and decreases gradually at larger curvatures, owing to the angular adjustment of proteins. When an isotropic bending energy component, F_pi, is added with a relative strength of 10% (κ_pi/κ_p = 0.1 and C_0a = 0), the density profile of ϕ_cy approaches a mirror symmetric shape (see the green line in Fig. 5(a)). Some amphipathic peptides have a kink structure, which allows significant bending. To mimic this behavior, a kink is introduced at 20% of the protein length from the protein end; at the kink, the protein bends laterally at an angle of π/4. Owing to the resulting asymmetry, the angular distribution becomes skewed, with the highest peak appearing at θ_pc < 0 and θ_pc > 0 for the curvature ranges 1 < 1/R_cyC_p < 2 and 1/R_cyC_p > 2, respectively (see the blue lines in Fig. 5).⁵⁶ A similar asymmetric angular distribution was reported in molecular simulation.¹¹⁹ The above discussion focuses on the binding of rigid proteins; however, the deformation of the binding domains can modify the protein density as demonstrated in ref. 56.

Fig. 5 Binding of anisotropic proteins in the low-density limit with C_p²a_p = 0.2, κ_p = 50 k_BT, and κ_s = 0.⁵⁶ The red lines represent the data of a twofold rotationally symmetric protein (crescent-rod shape without kinks). The blue lines represent the data of an asymmetric protein, where the rod-shaped protein bends at a kink with an angle of π/4, positioned at 20% of the protein length from the end. The axis of the asymmetric protein is set to be θ_peak = 0 at 1/R_cy ≪ 1. The green line represents the data of the twofold rotationally symmetric protein with an isotropic segment of κ_pi/κ_p = 0.1. (a) Binding density ϕ_cy on a cylindrical membrane with respect to the density ϕ_flat on a flat membrane. (b) Peak position of the angle θ_pc. The solid and dashed lines represent the first and second peaks, respectively. The inset shows the schematics of the top and side views of proteins. (c) Distribution of the angle θ_pc. The solid and dashed lines represent the data for 1/R_cyC_p = 3 and 0.8, respectively.

3.3.2 High protein density. Next, we describe a mean-field theory^121,129 that accounts for orientation-dependent excluded area, in which Nascimentos' theory¹³⁰ for three-dimensional (3D) liquid-crystals is applied to the 2D membrane. Bound proteins are assumed to adopt an elliptical shape laterally on the membrane and can be aligned based on their inter-protein interactions and their preferred bending direction. The degree of orientational order S is given by S = 2〈s_p(θ_ps)〉, where s_p(θ_ps) = cos²(θ_ps) − 1/2 and θ_ps denotes the angles between the major protein axis and nematic orientation S. The protein area is defined as a_p = π[small script l]₁[small script l]₂/4, where [small script l]₁ are [small script l]₂ represent the lengths of the major and minor protein axes, respectively.

The free energy F_p of bound proteins is expressed as¹²¹

where

g = 1 − ϕ[b₀ − b₂Ss_p(θ_ps)]. (28)

Ψ and represent the symmetric and asymmetric components of the nematic tensor, respectively. The factor g accounts for the weight of the orientation-dependent excluded volume interaction, and Θ(x) denotes the unit step function. When two proteins are aligned parallel to each other, the excluded area A_exc between them is smaller compared to when they are oriented perpendicularly. This difference increases with increasing aspect ratio d_el = [small script l]₁/[small script l]₂. The area A_exc can be approximated as A_exc = [b₀ − b₂(cos²(θ_pp) − 1/2)]a_p/λ, where θ_pp is the angle between the major axes of two proteins, and λ represents the packing ratio. The maximum density is given by ϕ_max = 1/λ(b₀ − b₂/2).¹²⁹

For a flat membrane, proteins exhibit an isotropic orientation at low densities and a first-order transition to a nematic order at high densities owing to the orientation-dependent excluded volume interactions.¹²¹ In this review, we consider the anisotropic bending energy described by eqn (19) with κ_s = 0 for U_p. As the curvature 1/R_cy of a membrane tube increases, proteins tend to align in the azimuthal direction even in the dilute limit (see Fig. 5(c)), and the transition to the nematic state becomes continuous.

For narrow tubes with 1/R_cy > C_p, the preferred protein orientation tilts away from the azimuthal direction. At low ϕ, proteins tilted in both the left and right directions coexist equally (Fig. 5). However, at high protein densities, only one type of tilt direction dominates due to orientation-dependent excluded volume interactions. Thus, second-order and first-order transitions occur between these two states at medium and large curvatures, respectively.¹²⁹

This theory well reproduces the simulation results for crescent protein rods on a membrane tube, when the proteins are isotropically distributed.¹²⁹ However, the discrepancies arise when the proteins form a significant amount of clusters, since the current theory does not account for inter-protein attraction and assumes a homogeneous protein distribution.¹²⁹

3.4 Binding to tethered vesicle

A vesicle held by a micropipette forms a narrow membrane tube (tether) under a pulling force applied by optical tweezers, as illustrated in Fig. 6(a).^30,131–133 The tube radius can be controlled by adjusting the force strength. Tethered vesicles have been widely employed to study the curvature sensing of membrane proteins, including BAR proteins,^{67,69,72,79,134} ion channels,^135,136 GPCRs,⁶⁸ dynamin,¹³⁷ annexins,¹³⁸ and Ras proteins.¹³⁹

Fig. 6 Binding of I-BAR domain of IRSp53 to tethered vesicle. (a) Schematic of the experimental setup. A cylindrical membrane tube (tether) is extended by an optical trap and micropipette. (b) Confocal image of a vesicle with a tube of R_cy = 25 nm. Green and magenta indicate the fluorescence for I-BARs and lipids, respectively. (c) Protein density ϕ_cy in the tube normalized by that of the large spherical region ϕ_L. Circles, triangles, and squares indicate the experimental data of ϕ_cy/ϕ_L for ϕ_L = 0.01, 0.02, and 0.05, respectively. The solid lines are obtained using fitting by the anisotropic protein model with κ_p/k_BT = 82 and C_p = −0.047 nm⁻¹. The experimental data in (b) and (c) are reproduced from ref. 67. Licensed under CC BY (2015). The plot in (c) is reproduced from ref. 80 with permission from the Royal Society of Chemistry (2023).

Protein density in the membrane can be quantified using fluorescence intensity measurement, as shown in Fig. 6(b). For I-BAR domains, the density ratio ϕ_cy/ϕ_L between the membrane tube to large spherical regions reaches a peak at a tube curvature of approximately 0.05 nm⁻¹ and gradually decreases at larger curvature (see Fig. 6(c)).⁶⁷ This curvature dependence can be reproduced by the theory for elliptic proteins (eqn (25)–(28)) with κ_p/k_BT = 82, C_p = −0.047 nm⁻¹, and κ_s = 0.⁸⁰ Note that the theory for isotropic proteins (eqn (12) or (13)) can reproduce each curve using different κ_pi and C₀⁶⁷ but cannot simultaneously fit all three experimental curves.⁸⁰ This finding strongly supports the anisotropic nature of the curvature sensing in I-BAR domains. Therefore, the tethered vesicle serves as a valuable tool not only for investigating curvature sensing but also for estimating the bending properties of various membrane proteins. However, the dependence on the saddle-splay modulus (k₃ in eqn (23)) cannot be directly measured using the tethered vesicle, since K = 0 in the membrane tube. Instead, k₃ can be estimated by comparing curvature sensing data from the membrane tubes and spherical vesicles with the same mean curvature (see Fig. 3). Curvature sensing has been observed through protein binding to spherical vesicles with various sizes,^84,139,140 and the comparisons with the data in membrane tube were also reported in ref. 139 at small membrane curvatures. For the estimation of the protein properties, the sensing data at large curvatures are particularly significant, since the anisotropic characteristics become more pronounced in this regime (see Fig. 5(a)).

The force generated by the bending energy, while maintaining a fixed volume and surface area, is balanced with the external force f_ex at equilibrium. Under typical experimental conditions of the tethered vesicle, the membrane tube is extremely narrow, making the volume of the cylindrical tube negligible, as R_cy²L_cy/R_A³ ≪ 1.^81,141 In this limit condition, the vesicle shape is obtained from ∂F_cv1/∂L_cy = f_ex|_Acy of the cylindrical tube with A_cy = 2πR_cyL_cy.

For the binding of isotropic proteins, it is expressed as⁸¹

where ϕ_cy is given by eqn (8). For the bare membrane (ϕ_cy = 0), a linear relation is obtained between the force and the tube curvature as f_ex = 2πκ_d/R_cy, which is widely used to estimate the bending rigidity of the bare membrane.^29–32,131 The protein density ϕ_cy and the tube curvature 1/R_cy exhibit mirror and point symmetry with respect to f_ex/f₀ = 1, as shown in Fig. 7(a) and (b), respectively, where f₀ = 2πκ_dC_s represents the force at the sensing curvature C_s = 2H_s = κ_piC₀/(κ_pi − κ_d). At high μ, a first-order transition occurs twice symmetrically at both weak and strong forces f_ex (see the red and green curves in Fig. 7(a) and (b)). At the transition point, narrow and wide tubes with different protein densities can coexist.

Fig. 7 Protein binding to a membrane tube pulled by an external force f_ex. Protein density ϕ and tube curvature 1/R_cy are shown in (a), (c) and in (b), (d), respectively. (a) and (b) Isotropic proteins for μ/k_BT = −4, −2, −1, and 0 at κ_p/κ_d = 4 and a_pC₀² = 0.16. (c) and (d) Crescent elliptic proteins for μ/k_BT = −2, 0, and 2.5 at κ_p/k_BT = 60, κ_s = 0, d_el = 3, and a_pC_p² = 0.26. The solid lines represent equilibrium states. The black dashed lines represent metastable and free-energy barrier states (van der Waals loops). The isotropic proteins exhibit a first-order transition twice at large μ.⁸¹ In contrast, the anisotropic proteins exhibit it only once at a small curvature.¹²⁹

For the anisotropic proteins, the membrane curvature is obtained from the force balance as f_ex/2π = ∂f_p/∂(1/R_cy)|_ϕcy + κ_d/R_cy, where f_p is given by eqn (26).^80,129 The f_ex dependence curves of ϕ_cy and 1/R_cy are not symmetric, unlike for isotropic proteins (compare Fig. 7(c) and (d) with Fig. 7(a) and (b)). The density and curvature exhibit a weaker dependence on f_ex at f_ex > f₀ owing to the protein tilting in narrow tubes, where f₀ = 2πκ_dC_p. Consequently, at high μ, the first-order transition occurs only once in wide tubes. This transition has been experimentally observed, showing the coexistence of high and low I-BAR density regions within the same membrane tube in ref. 67. The sensing curvature of anisotropic proteins is influenced not only by C_p but also by the protein density, as shown in Fig. 7(c) and (d).¹²⁹

4 Curvature generation

4.1 Isotropic proteins

Curvature-inducing proteins alter the local membrane curvature, bringing it closer to their preferred curvatures. In the absence of constraints, the curvature H_g generated by isotropic proteins is determined by minimizing the free-energy, given by the condition dF_cv1/dH = 0 using eqn (5):⁶⁶where κ_dif represents the bending-rigidity difference as used in eqn (10). Since the proteins bend the underlying membrane together, H_g is smaller than the sensing curvature H_s and depends on the membrane rigidity κ_d, unlike the sensing curvature (see Fig. 3(b)). Moreover, H_g differs between spherical and cylindrical membranes at [small kappa, Greek, macron]_pi ≠ [small kappa, Greek, macron]_d (see Fig. 3(b)). In the presence of constraints, the membrane may bend to a lesser extent than this generation curvature, since the constraints can suppress the membrane deformation.

4.1.1 Budding and vesicle formation. As an example of constraints, we consider the budding of a vesicle induced by protein binding. Here, the volume and surface area of the vesicle is fixed (constrained) so that local membrane deformation maintains these constraints by entailing deformation in other membrane regions.

In living cells, spherical buds typically form during vesicle formation. In clathrins-mediated endocytosis, clathrins assemble on the membrane, forming spherical buds with diameters ranging from 20 to 200-nm.^4,142–144 Similarly, in the membrane trafficking between the endoplasmic reticulum and the Golgi apparatus, COPI and COPII coated vesicles with diameters ranging from 60 to 100-nm are generated through budding under typical conditions.^5,145,146 These proteins can be considered as laterally isotropic, and their budding processes have been theoretically analyzed using a spherical-cap geometry^147–149 and more detailed geometry.¹⁵⁰

The budding of a vesicle can be understood using the mean-field theory with simplified geometries.⁶⁶ A budded vesicle is modeled as small spheres connected to a large spherical membrane, as depicted in the inset of Fig. 8(c). Assuming that all buds have the same radius R_bud, the free energy minimum can be easily solved using eqn (7) for one degree of freedom, since the other two lengths can be determined by the area and volume constraints. A prolate vesicle can be modeled by a cylinder shape capped with two hemispheres. As the chemical potential μ increases, the protein density ϕ_bud in the buds increases greater than ϕ_L in the large spherical region, leading to the formation of a greater number of buds with a smaller radius (see Fig. 8). At a small spontaneous curvature (C₀R_A = 200), The number of buds increases continuously, whereas, at a large spontaneous curvature (C₀R_A = 300), a first-order transition occurs from a few buds with a large radius to many buds with a small radius, as shown in Fig. 8. Thus, many buds can suddenly form after a long incubation period at slightly higher than the transition point.

Fig. 8 Budding of a vesicle induced by the binding of proteins with a spontaneous curvature C₀ at V* = 0.95, κ_pi/κ_d = 4, and [small kappa, Greek, macron]_pi/[small kappa, Greek, macron]_d = 1.⁶⁶ (a) Protein density on the vesicle surface in the absence of the ADE energy. The solid and dashed lines represent the densities in the buds ϕ_bud and spherical vesicle region ϕ_L, respectively. The blue and red lines indicate the continuous and discontinuous transitions at C₀R_A = 200 and 300, respectively. (b) Number n_bud of buds. The blue and red lines represent the data at C₀R_A = 200 and 300, respectively, in the absence of the ADE energy (corresponding to the data shown in (a)). The green and magenta lines represent the data with the ADE energy in the presence and absence of protein insertion (the insertion area ratio of the protein γ_in = 0.02 and 0), respectively, at R_A/h = 5000. (c) Free energy profiles at μ/k_BT = −0.7, −0.65, and −0.6 (from top to bottom) at C₀R_A = 300 (corresponding to the red lines in (a)). Two minima for a few and many buds appear around the transition point.

This simplified geometrical framework can be easily applied to other shape transformations and is useful for investigating the effects of additional interactions. For instance, the ADE energy is incorporated into the budding process (see Fig. 8(b)). Initially, the ADE energy is considered to be relaxed in the prolate vesicle (ΔA = ΔA₀ in the prolate). When the bound proteins do not change ΔA₀, the ADE energy only slightly reduces the budding (see the magenta curve in Fig. 8(b)).⁶⁶ However, the insertion of hydrophobic segments into the membrane can modify ΔA₀. When the segments insert only the outer leaflet with the ratio γ_in of the inserted area (i.e., ), the budding can be promoted (see the green curve in Fig. 8(b)). The insertion can induce the budding even at C₀ = 0 through the protein binding to the large spherical region.

Lipid membranes supported on a solid substrate are widely used as model systems for biological membranes, providing a valuable platform to study both protein functions and membrane properties.^151–155 Boye and coworkers reported that the annexin proteins^64,156,157 can detach lipid membranes from the substrate.^158,159 Their observation revealed membrane rolling and budding from open edges, with variations depending on the types of annexins. The budding and vesicle formation observed in these experiments can be interpreted as the binding behavior of isotropic proteins. Fig. 9 shows the membrane detachment dynamics obtained by a meshless membrane simulation,¹⁶⁰ in which particles with a diameter of σ self-assemble into one-layer sheets in a fluid phase. The bound proteins (represented as red particles) induce membrane bending, counteracting the adhesion to the substrate, leading to the formation of small vesicles from the membrane edge.

Fig. 9 Sequential snapshots of membrane detachment from a substrate induced by the binding of isotropic proteins at C₀σ = 0.2, κ_pi/k_BT = 34, κ_d/k_BT = 16, and μ/k_BT = 5.¹⁶⁰ Detached membranes form small vesicles. A sliced snapshot from the side view is shown for the right bird's-eye view snapshot. The red and yellow spheres represent the membrane particles with and without the protein binding, respectively. In the side view, the light gray rectangle represents the substrate.

4.1.2 Phase separation. Proteins exhibit both direct and membrane-mediated interactions, and their assemblies often influence the membrane morphology. The curvature generated by proteins can drive phase separation, resulting in protein-rich and protein-poor membrane domains with distinct curvatures. The vesicle budding process described in the Section 4.1.1 represents an extreme case of phase separation, where protein-rich buds form in contrast to the protein-poor large spherical region.

Under conditions of high surface tension compared to the spontaneous curvature of the domain and the line tension of the domain boundary, curved domains do not fully close into spherical buds but instead adopt a spherical-cap shape. When these spherical-cap domains expand to cover most of the membrane surface, they organize into a hexagonal array, representing the closest packing configuration in 2D space, as shown in Fig. 10(a).⁷¹ As the binding chemical potential μ of proteins increases, the membrane undergoes a continuous transition from an unbound state to a hexagonal phase. This is followed by a first-order transition to the homogeneously bound phase, where the entire membrane becomes saturated with proteins.⁷¹

Fig. 10 Phase separation induced by binding of isotropic proteins. (a)–(c) Binding to upper and lower membrane surfaces at C₀σ = ±0.1, κ_pi/k_BT = 144, κ_d/k_BT = 16, μ_u/k_BT = 7.5, and μ_ff = μ_d − μ_u.¹⁶⁰ The red and green spheres represent membrane particles bound from the upper and lower surfaces, respectively. The yellow spheres represent unbound membrane particles. (a) Hexagonal pattern of the upper-bound domains in the unbound membrane at μ_d/k_BT = 4. Lower bound particles are negligible. (b) Kagome-lattice pattern at μ_d/k_BT = 6. The upper- and lower-bound domains form hexagonal and triangular shapes, respectively. (c) Checkerboard pattern at μ_d = μ_u. Both upper- and lower-bound domains form square shapes. (d) Beaded-necklace-shaped membrane tube induced by binding to the outer surface.⁸¹ The red and yellow spheres represent bound and unbound membrane particles, respectively.

When proteins bind to both membrane surfaces from the upper and lower buffers, the membrane can form both convex and concave domains, as shown in Fig. 10(b) and (c).^160,161 Under symmetric conditions, where the chemical potentials of the upper and lower surfaces are equal (μ_u = μ_d), the membrane exhibits distinct patterns depending on the chemical potential. At low chemical potentials, square domains arranged in a checkerboard pattern obtained, while at higher chemical potentials, striped patterns emerge. Small unbound membrane patches stabilize the vertices of the square domains (see Fig. 10(c)). When repulsive interactions are added between the unbound and bound membranes, these unbound patches expand and take on a square shape, and the bound domains adopt an octagonal shape, resembling the 4.8.8 tiling pattern.¹⁶¹ Under asymmetric conditions, where the chemical potential of the upper surface exceeds that of the lower surface (μ_u > μ_d), a kagome-lattice pattern can form. In this configuration, triangular concave domains are arranged within a hexagonal array of convex domains (see Fig. 10(b)). As the chemical-potential difference further increases, concave domains disappear and a hexagonal pattern of convex domains form (see Fig. 10(a)). Additionally, the transfer (flip–flop) of proteins between the two surfaces can be accounted for using the flip–flop chemical potential μ_ff. At thermal equilibrium (μ_ff = μ_d − μ_u), the flip–flop does not change the equilibrium behavior owing to the principle of detailed balance. However, under non-equilibrium conditions (μ_ff ≠ μ_d − μ_u), the ballistic motion of biphasic domains and time-irreversible fluctuations of patterns can be observed.¹⁶¹

Phase separation can also occur in both spherical and cylindrical membranes. In spherical vesicles, the formation of hexagonal arrays of concave domains has been theoretically investigated.¹⁶² In cylindrical membrane, a 1D periodic pattern can emerge, in which round bound and narrow straight unbound domains alternate in a beaded-necklace-like arrangement (see Fig. 10(d)).⁸¹

Even in the absence of spontaneous-curvature differences between bound and unbound membranes, attraction between bound membrane regions can arise due to hydrophobic mismatch of transmembrane proteins and Casimir-like interactions in rigid proteins. The height of the transmembrane proteins can differ from the thickness of surrounding membrane,^163–165 resulting in an effective attraction between proteins to reduce the hydrophobic mismatch.^166–169 In thermal equilibrium, the membrane height fluctuations follow the relation 〈|h_q|²〉 = k_BT/(γq² + κq⁴), where h_q represents the Fourier transform of the membrane height in the Monge representation.^170,171 Here, the surface tension γ corresponds to the mechanical frame tension conjugated to the projected membrane area.¹⁷² Rigid proteins with high bending rigidity κ_p suppress membrane fluctuations in their vicinity. As a result, protein assembly mitigates entropy loss, leading to a Casimir-like attractive interaction.^71,173 This interaction is expressed in the leading order as 6 k_BT(r_p/r)⁴, where r is the inter-protein distance and r_p represents the protein length. Consequently, the binding of rigid proteins induces a first-order transition between unbound and bound states.⁷¹ Additionally, Casimir-like interaction also arises between ligand–receptor pairs that connect adjacent membranes, effectively reducing the fluctuations in the membrane separation distance.^174,175

4.2 Anisotropic proteins

4.2.1 Interprotein interactions. For anisotropic proteins, excluded volume interactions are orientation-dependent, as discussed in Section 3.3.2. Membrane-mediated interactions also depend on the protein orientation.^124,176,177 In a tensionless membrane (γ = 0), the curvature-mediated interaction energy for an isolated protein pair can be expressed in the leading order as¹²⁴where θ₁ and θ₂ denote the angles between the bending axis of proteins 1 and 2 and the vector r₁₂ connecting their centers. Two rigid proteins with curvatures C_r1 and C_r2 and a length of r_p are modeled as point-like objects,^{124,178–180} which allows the derivation of eqn (31). Similar angular-dependent interactions have been reported by assuming elliptical¹⁷⁶ and circular protein shapes.¹⁷⁷

When two proteins bend the membrane in the same direction (C_r1C_r2 > 0), they exhibit an attractive interaction when oriented side-by-side (θ₁ = θ₂ = π/2) and a weaker repulsive interaction when aligned along the membrane axis (θ₁ = 0 or θ₂ = 0). In the side-by-side dimer configuration (i.e., θ₁ = θ₂ = π/2), the membrane experiences reduced deformation. This bending-energy reduction is the origin of this attraction. Conversely, when the proteins bend the membrane in the opposite directions (C_r1C_r2 < 0), the interactions are reversed. In this case, the proteins exhibit weak attraction when aligned along the membrane axis (θ₁ = 0 or θ₂ = 0) and repulsion when positioned side-by-side (θ₁ = θ₂ = π/2). Therefore, proteins with similar curvatures preferentially interact in a side-by-side configuration, whereas proteins with opposite curvatures prefer tip-to-tip alignment. These interactions have been quantitatively confirmed through the meshless membrane simulations.¹²⁴ Furthermore, the Casimir-like interaction between straight rods exhibits a different angular dependence but decay over a shorter range, proportional to r₁₂⁻⁴.^181,182

For positive surface tensions (γ > 0), the bending energy dominates interactions on length scales shorter than whereas surface tension effects become dominant at length scale greater than r_ten. As a result, the interaction energy changes from a bending-dominant regime to a tension-dominant regime at approximately r₁₂ ≈ 3r_ten:¹²⁴

where

In some coarse-grained simulations, the tip-to-tip assembly of crescent proteins on membranes has been reported.^122,123 In these systems, proteins sink into the bound membrane by a strong protein-membrane attraction, resulting in a strongly negative spontaneous curvature perpendicular to the major axis of the crescent proteins. Consequently, the protein bending axis is perpendicular to the major axis, meaning that tip-to-tip alignment, from the perspective of the protein's shape, corresponds to side-to-side alignment when viewed from the bending axis.¹²⁴

4.2.2 Tubulation. The binding of BAR superfamily proteins to the membrane induces the formation of tubules. Tubulation from liposomes has been observed in in vitro experiments (see Fig. 11(a)).^{73,105–107,109} In living cells, different types of BAR proteins localize to tubular membranes in specific organelles and membrane regions.^1,3 Within these tubules, the helical assembly of BAR domains has been visualized using cryo-electron microscopy (EM), as shown in Fig. 11(b).^106–108

Fig. 11 Tubulation generated by BAR domains. (a) Confocal image of tubular invagination generated by the binding of I-BAR domains. Reproduced from ref. 109. Licensed under CC BY (2016). (b) N-BAR (amphiphysin/B1N1s)-coated tube with a diameter of 280 Å. 3D reconstruction from cryo-EM images. Reproduced from ref. 108. Licensed under CC BY (2015). (c) Tubulation simulated by a meshless membrane model.¹⁸³ The BAR domain and membrane beneath are modeled as a linear chain of (red and yellow) particles with two kink (light blue and yellow) particles for the molecular chirality. In the upper panel, a protein rod is extracted to show the structure. The gray spheres represent the bare membrane particles.

Tubulation and other membrane deformations have been realized using meshless membrane simulations (Fig. 11–14). In these simulations, the protein rods are modeled as linear chains consisting of ten membrane particles, with or without two kink particles to account for chirality, as shown in Fig. 11(c). The rod curvature C_rodr_rod ≃ 3 corresponds to that of BAR-PH domains,¹⁸⁴ where r_rod is the rod length. Additionally, excluded polymer chains, each containing n_poly Kuhn segments to represent IDP domains, are incorporated, as shown in Fig. 13(a). Tubulation with a helical protein assembly can be effectively reproduced using meshless simulations of chiral protein rods (see Fig. 11(c)).¹⁸³ While tubulation can also be induced by the achiral protein rods, the chirality has been shown to enhance the tubulation process.¹⁸³

Fig. 12 Tubulation from a flat membrane by crescent-rod proteins at ϕ_rod = 0.4 and C_rodr_rod = 4.¹²⁰ The proteins have the spontaneous curvatures C_rod and C_s along the protein axis and perpendicular (side) direction, respectively, as shown at the middle bottom in (a). The initial state is an equilibrium state at C_rod = C_s = 0, and the rod curvatures are tuned at t = 0. (a) The left panels show the sequential snapshots at t/τ = 0, 12.5, and 100 for a positive side curvature (C_sr_rod = 1). The right panels show the sequential snapshots at t/τ = 10, 100, and 200 for a negative side curvature (C_sr_rod = −1). The chains of spheres (upper and lower half surfaces are in red and yellow, respectively) represent the protein rods, and the gray spheres represent the bare membrane particles. (b) Time evolution of mean cluster height 〈z_cl²〉^1/2 normalized by the protein length r_rod. The solid lines represent the data at the surface tension γr_rod²k_BT = 0, 6.25, and 12.5 for C_sr_rod = − 1. The dashed line represents the data at γ = 0 for C_sr_rod = 1. Reproduced from ref. 120. Licensed under CC BY (2016).

Fig. 13 Tubulation and budding induced by crescent-rod proteins with anchoring excluded-volume chains at ϕ_rod = 0.24.¹⁸⁵ (a) A protein comprising a crescent rod with two (light blue) kink particles (for chirality) and two excluded-volume chains of n_poly particles, as a model of BAR proteins. (b) An array of short tubules at C_rodr_rod = 3 and n_poly = 25. (c) Long tubules at C_rodr_rod = 3 and n_poly = 100. (d) Ellipsoidal buds at C_rod = 0 and n_poly = 50. (e) Shish-kebab-shaped tubules at C_rodr_rod = −3 and n_poly = 50.

Fig. 14 Snapshots of vesicles with crescent protein rods. (a)–(d) A single type of protein is bound. Here, a protein is represented by a linear chain of ten spheres, whose upper and lower halves are in red and yellow, respectively. Unbound membrane particles are displayed in transparent gray. (a) Disk-shaped vesicle at ϕ = 0.167 and C_rodr_rod = 2.5. The proteins are in the disk edge.^78,186 (b) Linear protein assembly at ϕ = 0.167 and C_rodr_rod = 3.75.^78,186 (c) Tetrahedral vesicle at ϕ = 0.4 and C_rodr_rod = 2.5.⁷⁸ (d) High-genus vesicle at ϕ = 0.8 and C_rodr_rod = 4.¹⁸⁷ (e) and (f) Two types of proteins are bound with the densities ϕ₁ = ϕ₂ = 0.15.¹²⁴ Two types of proteins are displayed in red and yellow and in blue and green, respectively. (e) Disk-shaped vesicle at C_rod1r_rod = 4 and C_rod2r_rod = 2. The proteins are phase-separated in the disk edge. (f) Vesicle with bumps at C_rod1r_rod = 3 and C_rod2r_rod = −3. The linear protein assemblies with opposite curvatures are alternately aligned side-by-side.

Fig. 12 shows the tubulation dynamics of the achiral straight crescent rods.¹²⁰ The same type of protein rods exhibit a membrane-mediated attractive interaction when aligned side-by-side, as discussed in Section 4.2.1. Consequently, these protein rods initially form linear assemblies perpendicular to their axis. Over time, the contacts of these assemblies lead to the development of a network structure at a sufficiently high protein densities. Eventually, tubules protrude from the network (see Fig. 12(a)). The stability of this network structure is influenced by the side curvature C_s of the proteins and the membrane surface tension γ. A negative side curvature C_s reduces the bending energy at network branch points, leading to slower tubulation compared to the case where C_s > 0 (compare the dashed and solid lines at γ = 0 in Fig. 12(b)). Since tubulation results in a reduction of the projected membrane surface area, increasing membrane tension γ inhibits tubulation (see three solid lines in Fig. 12(b)).^73,183

The addition of the IDP domains can either promote or suppress tubulation, depending on the conditions.¹⁸⁵ For a short IDP with n_poly = 25, the tubulation dynamics slow down and become trapped in a short-tubule array, as shown in Fig. 13(b). In this case, the crowded IDP domains induce repulsion between tubules, preventing their fusion. Conversely, when n_poly = 100, the IDP chains extend beyond the mean distance between tubules, allowing fusion and promoting tubule elongation in the vertical direction (see Fig. 13(c)). Thus, interactions between IDP chains and membranes enhance tubulation, while interactions between the IDP chains of neighboring tubules slow it down. In the absence of spontaneous curvature in the binding domains, IDP domains facilitate the formation of ellipsoidal buds, since the IDP chains gain more conformational entropy in vertically elongated shapes (see Fig. 13(d)). When IDPs are introduced to negatively bent crescent rods – where the binding domain and IDPs exhibit the opposite spontaneous curvatures – periodically bumped tubules are formed (see Fig. 13(e)). For short IDP chains, the proteins assemble into a network structure, resembling Fig. 12(a), on the membrane. This assembly causes the membrane to become rugged due to the bumped assemblies. Notably, a similar rugged vesicle has been observed in experiments involving a chimeric protein composed of I-BAR and IDP domains.⁸⁵

4.2.3 Other membrane deformations. Fig. 14 shows vesicles deformed by the crescent protein rods. In vesicles and membrane tubes, protein assembly occurs in two distinct steps as the rod curvature C_rod increases.^78,186 At low C_rod, the proteins are randomly distributed. As C_rod increases to an intermediate level, the vesicle deforms into a disk-like shape, with proteins concentrating at the disk edge (see Fig. 14(a)). At high C_rod, proteins form an arc-shaped linear assembly, resulting in vesicle with flat disk and spherical regions (see Fig. 14(b)). In membrane tubes, proteins initially assemble in the azimuthal direction, causing the membrane to adopt an elliptic shape at a medium C_rod. As C_rod further increases, proteins also assemble along the tube axis. These assembly processes occur continuously, since each transformation progresses in a 1D manner.

At high C_rod and increasing protein density, the length of the protein assembly exceeds the edge length of the disk-shaped vesicle. Initially, the vesicle elongates into an elongated elliptical shape, eventually, forming polyhedral structures, such as a tetrahedral vesicle shown in Fig. 14(c). In membrane tubes, this process results in polygonal deformations, with proteins assembling along the edge lines of the polygon vertices.^70,78 Unlike the continuous transition described earlier, the transformations between polyhedral vesicles and between polygonal tubes are discontinuous.⁷⁸ Notably, similar triangular membrane tubes have been observed in the inner mitochondrial membranes of astrocytes.^188,189 At high C_rod and protein density, excessive protein-induce stress can lead to membrane rupture, giving rise to high-genus vesicles (see Fig. 14(d)).^187,190

When multiple types of proteins bind to a membrane, differences in their preferred curvatures can induce phase separation.^{124,191–193} When two types of proteins exhibit positive curvatures with different magnitudes, they can segregate into regions of large and small curvatures. In the case of a disk-shaped vesicle, proteins with larger curvature preferentially assemble at the corners of the triangular disk (see Fig. 14(e)). Conversely, when two types of proteins possess opposite curvatures, their 1D assemblies align alternately in a side-by-side arrangement, forming periodic bumps (see Fig. 14(f)).¹²⁴ Within this alternating pattern, the different proteins establish tip-to-tip contact, which is consistent with the attractive interactions in the tip-to-tip direction described in Section 4.2.1. Notably, this alternating assembly can also occur in flat membranes; however, it is disassembled under high surface tension.¹²⁴

Simulations showed that identical protein rods formed 1D linear assemblies through membrane-mediated interactions. The introduction of direct inter-protein interactions can modify the assemblies. The formation of helical tubular assemblies is further enhanced by direct attraction.¹⁸³ Specific types of direct interactions may be necessary to accurately describe the assemblies of certain proteins. The endosomal sorting complex required for transport (ESCRT) forms a distinctive assembly, characterized by a spiral-spring-like structure on flat membranes and a helical tube configuration on cylindrical membranes.^194–197 This spiral assembly is involved in endosomal fission. In dynamically triangulated membrane simulations,^{192,193,198,199} proteins are often represented as point-like inclusions with orientational degrees of freedom. In their models, protein interactions are governed by an orientation-dependent yet laterally isotropic potential. As a results, when the orientations and the distance between two proteins are fixed, the interaction energy remains identical for both side-by-side and tip-to-tip alignments. Owing to the attractive nature of this potential in both lateral directions, the resulting protein assemblies exhibit a thickness of a few proteins rather than forming a strict single-layer 1D structure.

4.3 Adhesion of nanoparticles

During phagocytosis, large objects, such as viruses and cell debris, are engulfed by the plasma membrane and internalized into the cell. The engulfment of colloidal nanoparticles has been extensively studied as a model system for phagocytosis, and nanoparticles are also widely considered as the carriers for drug delivery.^200–203 Unlike curvature-inducing proteins, an adhesive spherical nanoparticle can become fully wrapped by the membrane;^204–207 however, as surface tension increases, the membrane undergoes a first-order transition to a partially wrapped state.²⁰⁸ Similarly, liquid droplets can also be wrapped by the membrane, but in contrast, the partially wrapped droplets deform to satisfy the wetting conditions at the contact lines.^209,210 For non-spherical particles, the wrapping process may be accompanied by changes in particle orientation.^204,211,212

Nanoparticles exhibit membrane-mediated interactions, similar to those observed in membrane proteins.^204,213,214 Nanoparticles can induce the formation of membrane tubules, wrapping the nanoparticle assembly.²¹⁵ Simulations of nanoparticles with crescent²¹⁶ and hinge-like²¹⁷ shapes have been conducted as model systems for protein binding, revealing orientational assemblies analogous to those formed by anisotropic proteins. Note that these nanoparticles have negative spontaneous curvatures along their minor axes due to their rounded shapes.

5 Summary and outlook

This review examined the curvature-sensing and generation mechanisms of membrane proteins. Laterally isotropic proteins are capable of sensing both the mean and Gaussian curvatures of membranes, with their curvature dependence well described by the mean-field theory. The IDP chains increase the bending rigidity and spontaneous curvature of membranes, while decreasing the saddle-splay modulus. The binding of isotropic proteins can lead to the formation of spherical buds and periodic patterns, such as hexagonal, kagome-lattice, checkerboard arrangements, and beaded necklace tubes. The curvatures generated by proteins play a crucial role in stabilizing these phase-separated patterns. Additionally, the insertion of hydrophobic segments can modify the area difference between the two leaflets of the bilayer within the ADE model, ultimately inducing membrane budding.

The binding behavior of anisotropic proteins, such as those from the BAR superfamily proteins, depends not only on the membrane curvatures but also on protein orientations. Orientation-dependent excluded-volume interactions can drive an isotropic-to-nematic transition among the proteins. In the dilute limit, an isolated protein preferentially binds to wide cylindrical membrane tubes with its orientation aligned along the azimuthal or axial directions, whereas it binds to narrow tubes with two distinct tilted orientations. As protein density increases, these proteins undergo the first-order and second-order transitions from a state characterized by the coexistence of two tilt angles to an ordered phase with a single orientation angle, depending on the membrane curvature.

Anisotropic proteins are also capable of driving tubulation. Protein chirality enhances tubulation, whereas negative side curvature and positive surface tension counteract it. The IDP domains of BAR proteins promote tubulation while simultaneously inhibiting tubule fusion, leading to either accelerated or decelerated tubulation dynamics depending on the condition. Furthermore, anisotropic proteins can facilitate the formation of disk-shaped and polyhedral vesicles, polygonal tubes, and periodically bumped membranes.

For a quantitative understanding of the curvature sensing and generation, accurate estimation of protein bending properties is essential. This review described the estimation of bending properties of I-BAR domains through curvature-sensing studies using tethered vesicles. The same approach can be extended to other curvature-inducing proteins. To analyze the effects of Gaussian curvature, comparisons between cylindrical and spherical membranes with equivalent mean curvature are particularly important, especially at large curvatures. Additionally, the asymmetric protein shapes of proteins can be assessed by examining their orientation distributions in cylindrical and buckled membranes. Molecular dynamics simulations of proteins on a buckled membrane^119,218 provide valuable insights into their curvature-sensing properties and behavior.

Data availability

No primary research results, software or code have been included as part of this review.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP24K06973.

Notes and references

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