# A Rationale for Mesoscopic Domain Formation in Biomembranes

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## Abstract

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## 1. Introduction

## 2. In Thermodynamic Equilibrium

#### 2.1. Weak-Segregation Limit in the Vicinity of a Critical Point

#### 2.1.1. Curvature-Composition Coupling in Planar Membranes

- (a)
- The difference in lipid composition between both leaflets is important in cellular membranes, and it is maintained by the active cell metabolism. It can lead to bilayer spontaneous curvature if both leaflets conspire in this direction, because the bilayer curvature results form the difference in the spontaneous curvature of the monolayers [12]. The spontaneous curvatures of the main lipids found in plasma membranes are listed in [71] and they can be as large as $0.3$ nm${}^{-1}$ for cholesterol or 1,2-Dioleoyl-sn-glycero-3-phosphoethanolamine (DOPE). This is global on the whole membrane, but it can be accentuated locally due to the membrane lateral heterogeneity. For example, it has been shown on the basis of coarse-grained molecular dynamics simulations that mean curvatures of about $0.1$ nm${}^{-1}$ can be attained in asymmetric membranes containing separated Lo and Ld phases on one leaflet and pure unsaturated lipid on the other leaflet [72].
- (b)
- The difference in the aqueous solution composition on the two sides of the membrane is maintained by the cell [1]. As explained by Lipowsky in 2013, a difference of solute concentrations, including ions and small molecules, generically leads to spontaneous curvature when they adsorb onto the membrane surface, for purely entropic causes [73]. The membrane “bends away from the exterior compartment if the concentration ${c}_{\mathrm{ex}}$ in this compartment exceeds the concentration ${c}_{\mathrm{in}}$ in the interior compartment”. For a single solute with different concentrations across the membrane, the spontaneous curvature is given by$${C}_{\mathrm{sp}}=\frac{{k}_{\mathrm{B}}T}{4\kappa}\phantom{\rule{0.166667em}{0ex}}\ell \phantom{\rule{0.166667em}{0ex}}{\mathsf{\Gamma}}_{\mathrm{max}}\phantom{\rule{0.166667em}{0ex}}\frac{{c}_{\mathrm{ex}}-{c}_{\mathrm{in}}}{{K}_{d}},$$
- (c)
- The area difference between both leaflets can also lead to global spontaneous curvature. For instance, an area difference of $\sim 10$% leads a to spontaneous curvature ${C}_{\mathrm{sp}}\sim {10}^{-2}$ nm${}^{-1}$ [71]. This is the keystone of the area-difference-elasticity (ADE) model that has been developed to explain the rich shape variability of homogeneous lipid vesicles, in particular in function of their reduced volume v [12,75].

#### 2.1.2. Bending Modulus-Composition Coupling in Planar Membranes

#### 2.1.3. Vesicles

#### 2.2. Strong Segregation Limit

#### 2.2.1. Domain Buckling Induced by Line Tension—Spontaneous Symmetry Breaking

- Planar membranes—We begin with the simplest form of this mechanism, as proposed in planar geometry by Lipowsky in 1992 [110]. We consider a single membrane Lo domain (denoted $\beta $-phase in this work) in a large planar Ld membrane ($\alpha $-phase). Well below the demixing temperature, the boundary shape is close to a circle to minimize the interfacial energy. The total domain area is denoted by ${A}_{\beta}\equiv \pi {L}^{2}$ (L is its radius in the membrane plane). Lipowsky first assumes that the surface tension is vanishingly small ($\sigma =0$). If it buds in the third dimension, the domain adopts the shape of a spherical cap supported by a sphere of radius R, while the surrounding membrane remains flat (Figure 4a). The interface is now a circle of radius $N\le L$. Adopting a mechanical approach where fluctuations are ignored, the total elastic energy ${E}_{\mathrm{el}}$ of the domain is given by the sum of two antagonist contributions: the boundary line-energy ${E}_{\mathrm{bound}}=2\pi N\lambda $ that is proportional to the domain boundary length and tends to minimize it (by protruding in the third dimension); and the elastic Helfrich energy ${H}_{\mathrm{Helfrich}}$ which disfavors bending. For a fixed domain area ${A}_{\beta}$ the value of the cap radius R is obtained by minimizing ${E}_{\mathrm{el}}\left(R\right)$. A natural length-scale ${\xi}_{I}=\kappa /\lambda $ can be introduced, called the “invagination length”. If ${\kappa}_{\beta}\sim 100{k}_{\mathrm{B}}T$ for the Lo phase [106] and $\lambda \approx 1$ pN far from the critical point, then ${\xi}_{I}\approx 400$ nm. When getting closer to the critical point, $\lambda $ decreases as ${({T}_{c}-T)}^{\nu}$ with $\nu $ a universal critical exponent equal to 1 in 2D biphasic systems in the 2D Ising universality class [15,110] and ${\xi}_{I}$ grows. We shall come back to these values later in the Discussion Section.Lipowsky shows that if $L<4{\xi}_{I}$, then the optimal geometry is a flat domain ($N=L$); conversely, if $L>4{\xi}_{I}$, it is a complete sphere ($N=0$), protruding upward or downward with equal probability. Differently said, this simple model without surface tension proposes that above a critical line tension$${\lambda}_{c}=4\frac{{\kappa}_{\beta}}{L},$$
- Additional role of surface tension—The case where the surface tension is finite, $\sigma >0$, has been explored in detail in [107]. As bending stiffness, membrane tension applied in the membrane plane favors flat domains and comes in opposition to interfacial energy minimization. In this case also, and without necessarily appealing to spontaneous curvature, incomplete budding occurs above a critical line tension, through the spontaneous symmetry breaking principle (Figure 4c). The transition from flat to dimpled domains is now continuous whereas it was discontinuous without tension. More quantitatively, it is proven in this work that the critical line tension is given by ${\lambda}_{c}\simeq 8{\kappa}_{\beta}/L$ in the limiting case where the domain area ${A}_{\beta}=\pi {L}^{2}\ll {\kappa}_{\beta}/\sigma $. Here ${\kappa}_{\beta}$ is the domain stiffness, which can be different from the surrounding membrane one, ${\kappa}_{\alpha}$. Coming back to the notations used in the paragraph just above, this condition reads $L=8{\xi}_{I}$ at the critical point, which is twice the transition value found when $\sigma =0$. This means that in the interval $4{\xi}_{I}\le L\le 8{\xi}_{I}$, budding is energetically favored when $\sigma =0$, but becomes less stable than the flat geometry as soon as $\sigma $ is positive, even if small.Furthermore, just above this critical value, the contact angle $\u03f5$ at the domain boundary scales as $\left|\u03f5\right|\propto \sqrt{\lambda /{\lambda}_{c}-1}$. The Lo domain continuously but rapidly deviates from the flat state. By up-down symmetry, the domain is equally likely to bud upward ($\u03f5>0$) or downward ($\u03f5<0$). When ${C}_{\mathrm{sp}}\ne 0$, this symmetry is again explicitly broken, and phase diagrams can also be inferred [107].
- Vesicles—Jülicher and Lipowsky addressed the same question in the case of biphasic vesicles with spherical topology [111]. As above, different situations exist, but the up-down symmetry (more precisely the exterior/interior symmetry in this case) is explicitly broken on a vesicle. As stressed in the field-theoretic approaches presented in Section 2.1.3, a new ingredient can come into play here, namely the conservation of the volume V enclosed by the vesicle, or equivalently the pressure jump across the membrane, $\mathsf{\Delta}p$, which can be controlled through the osmotic pressure difference. The control parameter is, e.g., the reduced volume $v\equiv 6\sqrt{\pi}V/{A}^{3/2}\le 1$, measuring the deviation to a sphere (for which $v=1$). If two domains coexist as above, describing the membrane through an elastic continuum theory, the minimization of the total energy provides the so-called “shape equations”, from which the equilibrium vesicle shape under the relevant constraints is derived. In particular, it depends on the relative area fractions and on the different parameters (bending moduli $\kappa $, saddle-splay moduli ${\kappa}_{G}$, spontaneous curvature ${C}_{\mathrm{sp}}$), which can in principle be different in the two phases. Indeed, even though it is not a pre-requisite, the difference between the bending moduli of the two phases now likely plays a role, contrary to the planar case, because both phases are bent in this geometry.A rich phase diagram can be computed by minimizing the membrane energy, still neglecting thermal fluctuations. In this case as well, budding can be incomplete or complete, a closed vesicle then being connected to the main vesicle through an infinitesimal “neck”. However, a strong volume constraint $v\simeq 1$ (or equivalently large $\sigma $), where the shape is quasi-spherical, can act against the budding process but does not, in general, suppress it. The reader can refer to [111] for further details. These results have been confirmed by numerical coarse-grained modeling (4-bead lipids and explicit solvent) based on dissipative particle dynamics, where both area and volume are conserved [112].
- Experiments—Fluorescence microscopy experiments [105,106,113] have later validated this theoretical approach on free-floating giant unilamellar vesicles (GUV) made of ternary mixtures of saturated lipids, unsaturated lipids and cholesterol, well below the demixing temperature, which display separated Lo and Ld phases (Figure 4d). In reference [114], the reduced volume v of GUVs made of a DPPC/DOPC/cholesterol mixture is controlled by varying the osmotic pressure. If one starts from a spherical vesicle, domains bud (inward or outward according to the experimental conditions) when the enclosed volume decreases. Following these original studies, a series of papers studied the experimental counterpart of these theoretically predicted circular, budded Lo domains and established phase diagrams [63,107,115]. When the cholesterol concentration was increased above $\approx 35$%, a reversal phenomenon was observed, now with Ld domains in a Lo continuous background. The domain sizes were typically observed to be in the micron scale. We have previously explained that if $\sigma \ne 0$ [105], then the critical radius L above which domains buckles is $8{\xi}_{I}\sim $ few $\mathsf{\mu}$m with the above value of ${\xi}_{I}\approx 400$ nm. Experiments and theory are compatible. Even though in a less evident manner, AFM experiments also suggest that budding exists in planar geometry [25], as predicted by theoretical approaches in the relevant regimes of parameters.
- Elastic interaction between budded domains—In these experiments, it is also observed that domains sometimes coalesce [105,115] but that this process is very slow and does not follow the usual laws of coarsening [116]. The reason is that budded domains repel each other when they come in close proximity because they deform the elastic membrane, in an enhanced way if they are very close. This repulsion has even be very well quantified experimentally [63,107,115] and shown to be compatible with theoretical predictions. A supposedly metastable configuration is then observed with long but finite lifetime. After several hours, all Lo domains eventually coalesce and one ends with a complete macro-phase separation. Note also that coarsening is not always trapped and that the existence of normal coarsening has been correlated to a vesicle reduced volume v very close to 1 [115]. Indeed, budding requires excess area that is only available if the vesicle is at least slightly deflated.As a matter of fact, the complete proposed scenario is as follows: after quenching below the demixing temperature and once domain have nucleated, normal coarsening is initiated, with small but growing nanoscopic domains. Being small, these domains are flat as demonstrated above [63]. When their size reaches the critical value, all these domains suddenly buckle and coarsening is then trapped in the metastable state [107,115]. The lateral organization of domains observed on phase-separated Sphingomyelin(SM)/DOPC/cholesterol vesicles in [117] has been attributed to this inter-domain repulsion, and the force between domain measured. Strong slowing-down of domain coarsening observed in DPPC/DOPC/cholesterol GUVs [118] was also attributed to budding, even though the inter-bud repulsion was not explicitly appealed to in this work. In contrast, when budding is avoided on sufficiently taut vesicles, no slowing-down is observed with respect to the expected dynamical exponent [119].

#### 2.2.2. Competing Interactions: Phase-Dependent Bending Modulus

#### 2.3. Competing Interactions: Spontaneous Curvature Induced by Membrane Inclusions

#### 2.3.1. Inter-Protein Short-Range Forces

- Electrostatic, van der Waals and hydrogen-bond interactions—Polar and charged amino acids at their surface can interact when two proteins come in close proximity. The Debye length ${\lambda}_{D}\sim 1$ nm in water at physiological salt condition [2] sets the typical range above which these interactions are screened. Inside the apolar hydrophobic membrane region where the dielectric constant is weaker, the range can be somewhat larger, of a few nanometers [133,134]. The range of van der Waals and hydrogen-bond interactions is also nanometric.
- Hydrophobic mismatch—Integral proteins have transmembrane domains consisting of alpha helices with hydrophobic amino-acid side chains, buried inside the hydrophobic core of the lipid membrane. Protein and membrane hydrophobic core thicknesses do not necessarily match. Since exposure of hydrophobic residues to the aqueous solvent is energetically unfavorable, the membrane must be deformed in the case of significant mismatch [12,135]. If two (or more) proteins are in proximity, the overall energy penalty depends on their distance d. As above, an effective force ensues (Figure 7b). It is attractive when both mismatches have the same sign and repulsive in the converse case. The energies at play go from a fraction of ${k}_{\mathrm{B}}T$ to several ${k}_{\mathrm{B}}T$, depending on the degree of hydrophobic mismatch, and the range of these forces is few nanometers [136,137,138,139]. It has been suggested that hydrophobic mismatch forces are not pairwise additive [140].
- Casimir interaction—This attractive interaction, of entropic origin, is named by extension of the Casimir interaction in quantum physics (the attraction between conducting plates mediated by quantum fluctuations in the electromagnetic field). Here it results from the transverse thermal fluctuations of the elastic membrane. The number of vibrational degrees of liberty of a membrane in which two (or more) inclusions are embedded depends on their mutual distance d. The potential of mean force thus depends on d, and has been shown to behave as $\sim -{k}_{\mathrm{B}}T/{d}^{4}$ in the case of vanishing membrane tension $\sigma $ [131,141]. The calculation can be extended to the case $\sigma >0$, where the interaction energy decays much faster with d, as $\sim -{k}_{\mathrm{B}}T/{d}^{8}$ when $d\gg \xi $, and as $\sim -{k}_{\mathrm{B}}Texp(-d/\xi )$ when $d\lesssim \xi $ [142,143].
- Depletion (or excluded-volume) forces—Attractive depletion forces (Figure 7a) are well characterized in soft condensed matter when large particles evolve among smaller ones, and play a role in physical biology (see [2] for example). In the present case, they are due to the 2D osmotic pressure laterally exerted by the surrounding lipids on large transmembrane proteins (larger than lipids). It should be far less pronounced for peptides. Roughly speaking, when two proteins are far away, the lateral osmotic pressure is isotropic and no net force ensues. When the relative distance becomes on the order of the lipid lateral size (<1 nm), the interval between the two inclusions tends to be depleted in lipids, and the pressure is not isotropic anymore. This tends to bring proteins closer when they are about a nanometer away [144,145]. The ensuing binding energy is on the ${k}_{\mathrm{B}}T$ range, even though the actual value depends on the model details.
- Lipid wetting—Some lipids are known to have a preferential affinity for given proteins species [39,146,147,148], in particular but not exclusively because they better match their hydrophobic length. Even above the phase-transition temperature, the protein can nucleates a small “halo” of such lipids, the range of which is on the order of magnitude of the composition correlation length ${\xi}_{\mathrm{OZ}}$ (see Section 2). This mechanism known as “wetting” [132,149,150] is reminiscent of the “lipid annulus” or “lipid shell” concepts that have become popular in the biophysical literature a dozen of years ago [40]. When two proteins approach close enough for their halos to overlap, they tend to assemble because it reduces the net interfacial energy. An effective attractive force ensues (Figure 7c). This nucleation mechanism can also promote the formation of a lipid halo of a thermodynamic phase that would be unstable in absence of the inclusion. A similar mechanism has been demonstrated to emerge in a very illustrative way [151]. In all cases, the range is set by the correlation length ${\xi}_{\mathrm{OZ}}$.This force is enhanced near a miscibility critical point because the composition correlation length ${\xi}_{\mathrm{OZ}}$ grows significantly. Exactly at the critical point, a long-range, power-law decrease of the potential of mean force at large inter-inclusion distance d has been predicted by a conformal field theory approach, with exponent $-1/4$, and confirmed by Monte Carlo simulations of the Ising model [152]. Coarse-grained molecular dynamics simulations on a model membrane and a phenomenological Ginzburg-Landau theory have explored the same mechanism in the case of peripheral proteins adsorbed onto the bilayer and interacting preferentially with one lipid species (among two). They drawn similar conclusions [147]. The binding energy at close range for two identical particles is also found in the ${k}_{\mathrm{B}}T$ range.Note that this mechanism is specific to the protein species and the lipids with which it preferentially interacts because the halos must be miscible if the interaction is attractive. In the case where they are immiscible, the force can even become repulsive instead [147,152]. Small alterations in lipid chemical structure can thus lead to dramatic changes in the membrane organization. This mechanism has been evidenced in model membranes [146].

#### 2.3.2. The Cluster Phase Scenario

#### 2.3.3. Spontaneous Curvature Can Play the Same Role as a Long-Range Repulsion

#### 2.3.4. Sources of (Local) Spontaneous Curvature ${C}_{\mathrm{sp}}\ne 0$

- The transmembrane part of an integral protein has no reason to be up-down symmetric, not least because the cytosolic and extracellular protein regions do not have the same biological function. This is either apparent in the molecular shape of transmembrane proteins or can be inferred from their behavior in biophysical experiments [71,163,169,170,171,172,173,174]. However, it seems difficult to infer the spontaneous curvature from the sole molecular shape displaying up-down symmetry breaking, for the reasons that we discuss now.
- Peripheral proteins naturally break the up-down symmetry [70,175,176], to a degree that depends in particular on the depth of penetration of the hydrophobic domain of the protein into the bilayer [71]. Numerical evidence can be found for example in Ref. [147], where the small shoulder on the interaction potential at intermediate range indicates a weak repulsion. More generally, anchored molecules can play the same role, as it was non-ambigously demonstrated in reference [86] on experimental proofs.
- The coupling between lipid composition and protein wetting by lipids is also a potential source of local curvature if a protein recruits different lipids in the two leaflets, themselves promoting markedly differential local curvature of the two leaflets.

#### 2.3.5. Diversity of Membrane Proteins and Biological Specialization of Clusters

#### 2.4. A Unifying Rationale: Up-Down Symmetry Breaking

## 3. Active and Out-of-Equilibrium Processes

#### 3.1. Models

- (i)
- Master Smoluchowski’s coagulation equation [57,196,199,201]. For example, Turner et al. [57] studied the coagulation equation$$\frac{\mathrm{d}{c}_{n}}{\mathrm{d}t}=\zeta \left(n\right)+\sum _{m=1}^{\infty}{k}_{n,m}{c}_{n+m}-{k}_{n,m}^{\prime}{c}_{n}{c}_{m}+\frac{1}{2}\sum _{m=1}^{n-1}{k}_{m,n-m}^{\prime}{c}_{n-m}{c}_{m}-{k}_{m,n-m}{c}_{n},$$
- (ii)
- Non-linear reaction-diffusion equations [56,197,198,200,202], which can also be seen as Cahn-Hilliard equations [15,116] suitably modified to take recycling into effect. For instance, in Ref [56], the Cahn-Hilliard equation$$\frac{\partial \varphi}{\partial t}=-\frac{\varphi -\overline{\varphi}}{\tau}+A{\nabla}^{2}\frac{\partial w}{\partial \varphi}$$

- The off-rates (from the membrane to the cytosol) can be size-dependent [190] or not [57,199]. In the former case, it means for example that endocytosis is able to extract patches from the membrane with a limited size set by the endocytosed vesicle typical size [201]. A “recycling correlation length” can also be introduced in the modified Cahn-Hilliard equations, mimicking the spatial range of recycling processes, i.e., the typical size of membrane patches recycled through vesicle traffic [197,202]. In the models of Refs. [56,57,196,198,200], only monomers are locally extracted from the membrane.
- The on-rates (from the cytosol to the membrane) are also size-dependent. Several models only inject monomers or tiny domains in the membrane [56,57,190,196,198,199,200,201] because they do not assume any pre-order in the exocytosed patches or because they assume direct exchange of monomers from the cytosol to the membrane, e.g., for peripheral proteins. Indeed, Foret argues that the traffic should be modeled differently for peripheral and transmembrane proteins [196], because the former are preferentially exchanged as monomers between the cytosol and the membrane, while the latter preferentially escape and join the membrane by endo- and exocytosis, respectively. Another approach assumes that domains with a characteristic size are directly injected in the membrane [197,202].
- Inside the membrane, two mechanisms control the domain dynamics: either the domains principally exchange matter through Ostwald ripening (exchange of monomers via the surrounding dilute “gas” phase [116,193], as illustrated in Figure 8b), see references [56,196,197,198,199,200,202], or through domain scission or fusion events, for all sizes [57,190] (Figure 8a).

#### 3.2. Results and Prospects

## 4. General Discussion and Conclusions

#### 4.1. A Variety of Mechanisms in Equilibrium in the Strong Segregation Limit

#### 4.2. Nanodomains and Critical Density Fluctuations

#### 4.3. Needed Theoretical Clarifications

#### 4.4. Needed New Experiments

#### 4.5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Membrane simulation snapshot. The blue lipids are 1,2-dipalmitoylphosphatidylcholine (DPPC), the yellow ones di-C16:2-C18:2 PC (DIPC) and cholesterol appears in green. The pink thick line is the 1D interface delimiting the liquid-ordered (Lo) phase (cholesterol rich, $\varphi \left(\mathbf{r}\right)\simeq 1$) and the liquid-disordered (Ld) phase (cholesterol poor, $\varphi \left(\mathbf{r}\right)\simeq 0$). This interface has an energy cost per unit length, homogeneous to a force $\lambda $, named the line tension. Although fluctuating, the membrane is globally planar and parallel to the plane $\left(xOy\right)$. The height of the membrane above this reference plane $\left(xOy\right)$ is measured by the height function $z=h\left(\mathbf{r}\right)$. The membrane is taut with surface tension $\sigma $ as illustrated by the four gray arrows parallel to the plane $\left(xOy\right)$. Adapted from an original membrane image generated by the MARTINI force field, reproduced with the courtesy of Matthieu Chavent. (

**b**) In the cluster-phase scenario, proteins are described as individual objects embedded in a continuous fluctuating 2D lipid mattress, also represented by a height function. They gather because of short-range attractive forces but long-range repulsion between clusters limits their growth. In the present case, each individual protein (in purple) locally imposes a spontaneous curvature to the membrane that is represented as an elastic sheet (in orange). In a taut membrane, an effective long-range repulsion between proteins ensues (see text). The units are arbitrary.

**Figure 2.**Examples of phase diagrams showing the formation of curvature-induced domains in planar membranes. Each diagram shows 3 phases: the macrophase separation (M), the modulated phase (or mesophase) (O), and the liquid phase which can be simple (L) or structured disordered (SD) with transient domains. (

**a**) Coupling constant between the two leaflets ${m}_{0}$ versus ${C}_{1}/{C}_{1}^{\ast}$ (see text) from the model by Gueguen et al. in which the two leaflets have the same composition but with different averaged area fractions. SD- (resp. SD+) corresponds to curved (resp. flat) transient domains. Here the bending modulus $\kappa $ does not depend on the phase state. Reproduced from [85], with permission of The European Physical Journal (EPJ), Copyright 2014. (

**b**) Temperature versus ${C}_{1}/{C}_{1}^{\ast}$ from the model by Shlomovitz and Schick in which each leaflet contains a different mixture of two lipids. (SDin) corresponds to curved transient domains in the inner leaflet only, and (SD) in both leaflets. Adapted from [79], with permission from Elsevier, Copyright 2013.

**Figure 3.**Examples of phase diagram showing the formation of curvature induced domains in vesicles. (

**a**) AT high T or low ${C}_{1}$, the liquid phase is homogeneous; numbered regions correspond to modulated phases with the number of the most stable ℓ-mode ($\ell =1$ to 5 here; $\ell =1$ corresponds to the macrophase separation). The superscript corresponds to the sign of ${C}_{1}$. The bending modulus $\kappa $ does not depend on the order parameter $\varphi $ in the model. In this case, ${C}_{0}=\mathsf{\Delta}p=0$. Adapted from [104]. (

**b**) Sketch of a quasi-spherical vesicle with two different types of lipids inducing either thicker (with a larger bending modulus ${\kappa}_{0}+{\kappa}_{1}$) or curved patches with a local spontaneous curvature ${C}_{0}+{C}_{1}$. (

**c**) Associated phase diagram in the $({C}_{1}R,{\kappa}_{1}/{\kappa}_{0})$ plane. The symbols have the same signification as in Figure 2a. (

**b**) and (

**c**) are reproduced from [85], with permission of The European Physical Journal (EPJ), Copyright 2014.

**Figure 4.**(

**a**) In the case of plane tensionless membranes, a first approximation consists of considering spherical caps of Lo phase (or $\beta $ phase, in red) of area ${A}_{\beta}=\pi {L}^{2}$ in an otherwise infinite Ld membrane [110]; the spherical cap radius is denoted by R and the radius of the circle delimiting the phase boundary is N. (

**b**) When the line tension $\lambda $ exceeds a critical value ${\lambda}_{c}$ (Equation (18)), the previously planar domain buckles to a complete sphere in the tensionless case. Owing to the spontaneous symmetry breaking principle, the domain is equally likely to buckle upward or downward (up-down symmetry). (

**c**) In a taut membrane with surface tension $\sigma $ budding is incomplete. Reproduced from [107] with permission from the Proceedings of the National Academy of Sciences USA, Copyright 2009. (

**d**) In the case of giant unilamellar vesicles (GUV), fluorescence microscopy experiments on ternary mixtures of 1,2-Dioleoyl-sn-glycero-3-phosphocholine (DOPC)/Sphingomyelin (SM)/cholesterol below the demixing temperature distinguishes the Lo and Ld phases appearing in red and green respectively. Mathematical modeling then enables the extraction of physical parameters such as bending moduli $\kappa $ and line tension $\lambda $ at the Lo/Ld interface. Reproduced from [106], with permission from the American Physical Society, Copyright 2008.

**Figure 5.**(

**a**) Numerical model of biphasic vesicle with Lo (in white) and Ld (in red) separated phases, sampled by Monte Carlo simulations. The ratio ${\kappa}_{\mathrm{Lo}}/{\kappa}_{\mathrm{Ld}}$ is indicated below each vesicle. Most curvature is accumulated in the more flexible Ld phase. Reproduced from [53], with permission from The Royal Society of Chemistry, Copyright 2011. (

**b**) Another numerical modeling of a biphasic vesicle with various Lo and Ld area fractions (Lo is black and Ld is white), showing patterned morphologies. Panel (

**a2**) is the same simulation snapshot as panel (

**a1**), but displays the mean curvature map instead (curvature units in the color scale on the right are in $\mathsf{\mu}$m${}^{-1}$), thus illustrating the coupling between the local composition and curvature. The most flexible Ld phase is also the most bent one. Reproduced from reference [123], with permission from the American Physical Society, Copyright 2013. In both panels (

**a**,

**b**), the spontaneous curvature ${C}_{\mathrm{sp}}$ has been set to 0 for both phases. (

**c**) Fluorescence microscopy images showing a large variety of patterns in the 4-component 1,2-distearoyl-sn-glycero-3-phosphocholine(DSPC)/1,2-Dioleoyl-sn-glycero-3-phosphocholine (DOPC)/1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC)/cholesterol vesicles, displaying coexistence of Lo (dark grey) and Ld (light gray) phases below the demixing temperature ${T}_{d}$. The vesicles either appear homogeneous or display roundish domains or labyrinthine phases according to the concentrations of the different lipids. The percentage indicated in each panel is the DOPC to DOPC+POPC molar ratio $\rho $. Scale bars: 10 $\mathsf{\mu}$m. Temperature: 23 ${}^{\circ}$C. Reproduced from [49], with permission from Elsevier, Copyright 2013.

**Figure 6.**From right to left, the vesicle radius of curvature (written above each snapshot) is increased while the total area of the patch is held fixed. Above a critical curvature, there is a transition from a Lo/Ld mesophase to a macrophase separation. Reproduced from [128], with permission from the American Physical Society, Copyright 2014.

**Figure 7.**Illustrations of some of the inter-protein forces (denoted by F on this figure) propagated by the lipid membrane. (

**a**) Depletion forces due to the lateral osmotic pressure exerted by lipids (red discs) on the proteins (black). The pressure becomes anisotropic when the distance between proteins is smaller than the lipid size. (

**b**) Hydrophobic mismatch forces resulting from the difference between the thickness of the membrane hydrophobic layer (the hydrocarbon chains) and the height of the protein hydrophobic cores (in gray) (

**c**) Wetting-induced interaction: when the membrane is made of a lipid mixture, a single protein (left panel) nucleates a “halo” of “wetting” lipids (in red). When two proteins get closer (right panel), their halos overlap, which reduces the interfacial energy between lipid species. A force ensues. (

**d**) Mutual interaction felt by two up-down non-symmetric inclusions (in blue) and propagated by the deformable elastic membrane. The asymmetry is schematized by the conical shape of inclusions. It is measured by the cone half-aperture angle $\theta $. (

**e**) Collective deformation of the membrane by an assembly of up-down non-symmetric inclusions (pink cones). Figures reproduced from [9], with permission from Elsevier, Copyright 2016.

**Figure 8.**Examples of recycling schemes. (

**a**) In the membrane, domains can undergo scission or fusion, whatever their size. Reproduced from [57], with permission from the American Physical Society, Copyright 2005. (

**b**) In this alternative scheme, clusters can only gain or lose proteins by exchange of monomers with the surrounding membrane (Ostwald ripening) because scission and fusion events are assumed to be rare. Reproduced from [199], with permission from the American Chemical Society, Copyright 2016. In both examples, monomers are injected into the membrane from the cytosol at a rate ${j}_{\mathrm{on}}$, either by exocytosis in (

**a**) or by a monomer flux from the cytosol in (

**b**). Multimers are internalized through endocytosis with a rate ${j}_{\mathrm{off}}$ that is independent of their size.

**Figure 9.**Examples of steady-state domain-size distributions $p\left(n\right)$. Here the domains are protein clusters. In these figures, their size is measured as the number n of proteins they contain, proportional to ${L}^{2}$ if L is the cluster radius. The units on the vertical axis are arbitrary. (

**a**) Bimodal distributions, where small oligomers coexist with domains of typical size ${n}^{\ast}$ (vertical dashed line for curve f). When going from curve “a” to curve “f”, the number of monomers in the membrane increases because more and more monomers are injected in the out-of-equilibrium membrane from the cytosol. Above a critical value, multimers nucleate and the distribution becomes bimodal as in curves “d”, “e” and “f”. The limit between monomodal and bimodal distributions is curve “c”. Adapted from [196]. (

**b**) Power-law distributions $p\left(n\right)\propto {n}^{-3/2}$ with a cut-off ${n}_{\mathrm{max}}$ (vertical dashed line) above which $p\left(k\right)$ decreases exponentially. Here the coalescence rate between two clusters is independent of their size. Clusters smaller than a size ${n}_{\mathrm{v}}$ are recycled entirely as a whole, while larger clusters are fragmented and lose an area ${n}_{\mathrm{v}}$ during a recycling event. The “whole cluster recycling” limit corresponds to ${n}_{\mathrm{v}}\to \infty $ and the “monomer recycling” limit to ${n}_{\mathrm{v}}=1$. Adapted from [201]. For both figures, see the cited references for more details on the recycling dynamics.

**Table 1.**Main notations used in this review, together with the sections where they are defined and useful references.

Notation | Name | Section Defined | References |
---|---|---|---|

Lo | Liquid-ordered lipid phase | 1 | [7,12,13] |

Ld | Liquid-disordered lipid phase | 1 | [7,12,13] |

$\sigma $ | Membrane surface tension (i.e., energy per unit area) | 2 | [2,11] |

$\kappa $ | Bending elastic modulus or curvature rigidity | 2 | [2,11,15] |

${\kappa}_{G}$ | Saddle-splay elastic modulus | 2 | [11,15] |

$\xi $ | Helfrich correlation length: $\xi =\sqrt{\kappa /\sigma}$ | 2.1.1 | – |

$h\left(\mathbf{r}\right)$ | Height function (in the Monge representation) | 2 | [11] |

$\varphi \left(\mathbf{r}\right)$ | Order parameter: local area fraction or phase state (e.g., Lo/Ld) | 2 | [12] |

H | Mean curvature ($H={\nabla}^{2}h/2$ in the Monge representation) | 2 | [2,11,15] |

K | Gaussian curvature | 2 | [2,11,15] |

v | Reduced volume of a vesicle of volume V and area A: $v=6\sqrt{\pi}V/{A}^{3/2}\le 1$ | 2.2.1 | [53] |

${C}_{\mathrm{sp}}$ | Spontaneous or preferred mean curvature | 2 | [11,15] |

$\lambda $ | Domain line tension (i.e., energy per unit length at a 1D phase boundary) | 2 | [18,48] |

${\xi}_{I}$ | Invagination length: ${\xi}_{I}=\kappa /\lambda $ | 2.2.1 | [48] |

$\mathsf{\Lambda}$ or $\kappa {C}_{1}$ | Coupling coefficient between $\varphi \left(\mathbf{r}\right)$ and H | 2 | [18,45] |

${T}_{c}$ | Critical temperature (at a miscibility critical point) | 2 | [11,12,17,18] |

${T}_{d}$ | Demixing or phase-separation temperature | 2 | [12,17,18] |

${\xi}_{\mathrm{OZ}}$ | Ornstein-Zernike composition correlation length | 2 | [15,54] |

${k}_{\mathrm{B}}T$ | Thermal energy $\simeq 4.3\times {10}^{-21}$ J at physiological temperature (37 ${}^{\circ}$C) | 1 | [2] |

$\mathsf{\Delta}p$ | Pressure jump across the membrane (for closed vesicles): $\mathsf{\Delta}p={p}_{\mathrm{int}}-{p}_{\mathrm{out}}$ | 2.1.3 | [55] |

$\tau $ | Recycling time (out-of-equilibrium membranes) | 3.1 | [56,57] |

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Destainville, N.; Manghi, M.; Cornet, J. A Rationale for Mesoscopic Domain Formation in Biomembranes. *Biomolecules* **2018**, *8*, 104.
https://doi.org/10.3390/biom8040104

**AMA Style**

Destainville N, Manghi M, Cornet J. A Rationale for Mesoscopic Domain Formation in Biomembranes. *Biomolecules*. 2018; 8(4):104.
https://doi.org/10.3390/biom8040104

**Chicago/Turabian Style**

Destainville, Nicolas, Manoel Manghi, and Julie Cornet. 2018. "A Rationale for Mesoscopic Domain Formation in Biomembranes" *Biomolecules* 8, no. 4: 104.
https://doi.org/10.3390/biom8040104