Modeling membrane-protein interactions

In order to alter and adjust the shape of the membrane, cells harness various mechanisms of curvature generation. Many of these curvature generation mechanisms rely on the interactions between peripheral membrane proteins, integral membrane proteins, and lipids in the bilayer membrane. One of the challenges in modeling these processes is identifying the suitable constitutive relationships that describe the membrane free energy that includes protein distribution and curvature generation capability. Here, we review some of the commonly used continuum elastic membrane models that have been developed for this purpose and discuss their applications. Finally, we address some fundamental challenges that future theoretical methods need to overcome in order to push the boundaries of current model applications.

The interplay between cellular membrane and membrane proteins is one of the major sources of curvature production in live cells.Membrane protein interactions result not only from those proteins that are integral to the membrane but also from those proteins, such as scaffolding molecules or GTPases that can attach and detach from the membrane surface locally in response to signaling events.[17,18,[24][25][26][27].Many different mechanisms have been proposed for how proteins can generate curvature of the membrane; for the purposes of theoretical modeling and capturing the key physical principles, the broadly accepted mechanisms can be grouped into two main categories; (i) the hydrophobic insertion mechanism, and (ii) coat proteins with hydrophilic domains [18,28,29].In the hydrophobic insertion mechanism, membrane bending occurs due to the change in the relative area of the two membrane leaflets, which happens due to partially embedded amphipathic helices of the protein domains [30,31].In contrast, when proteins are thought to coat the membrane, there is no insertion into lipid bilayer and proteins simply oligomerize along the membrane surface [32,33].In this case, it has been suggested that the steric pressure generated due to protein crowding and scaffolding drive the membrane deformation [34][35][36].
There are various methods to visualize membrane curvatures in situ or in reconstituted systems such as X-ray crystallography [48,49], nuclear magnetic resonance spectroscopy (NMR) [50,51], fluorescence microscopy [52,53], and electron microscopy (EM) [54,55].Use of these provide an opportunity for scientists to decipher vast amounts of information about the molecular machinery underlying the membrane shape transformations at high resolution.However, taking high resolution images is expensive and biological systems are very dynamic making it challenging to experimentally quantify the role of a specific component, e.g., membrane-protein interaction, in biological phenomena [56][57][58].The use of theoretical and computational approaches have became popular as complementary techniques to explore the mechanochemical aspects of membrane curvature generating mechanisms and enable us to identify some of the key underlying physics [59][60][61][62][63][64][65].

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In this review article, we mainly focus on continuum models for incorporating the effects of membrane proteins on lipid bilayers and cell membranes.In section 2, we briefly introduce the basic components of biological membranes and highlight their structure and functions.Next, in section 3 we present the two different computational approaches for modeling membrane-protein interaction -molecular dynamics versus continuum models.In section 4, we provide an overview of some of the popular continuum models for describing the constitutive relationships of the plasma membranes in contact with proteins.We conclude the review with a discussion on the challenges and possible future directions of the theoretical methods in section 5.

Biological membranes
Biological membranes (BMs) are fundamental architectures that form the outer boundary of living cells or compartments inside the cell.BMs act as semi-impermeable barriers that separate the cell contents from the extracellular environment while they also allow the vital materials to pass into or out of the cell [66,67].The main component of all biological membranes is a lipid bilayer.The lipid bilayer thickness is about 5-10 nm and is made of three primary lipids: phospholipids, cholesterol, and glycolipid molecules (see Fig. 2) [67][68][69].Proteins are the second major component of cell membranes in which the weight ratio of the lipids to membrane proteins can vary from 20% to 70%, depending on the cell type [67,70,71].
Proteins in cell membranes are classified into two categories; integral and peripheral proteins [72,73] (see Fig. 2).The third major component of BMs is carbohydrate molecules, which are found on the extracellular side of cell membranes.[74,75].
Carbohydrates are usually short and branched chains consisting of plain sugars, amino sugars, and acidic sugars.We briefly survey the different classes of membrane proteins, their functions, and their structures in cell membranes in what follows.

Integral proteins
Integral proteins are embedded permanently in the membrane by hydrophobic, electrostatic, and other non-covalent interactions [76,77].Therefore, removing integral proteins from lipid bilayer is only possible by the use of detergents or nonpolar solvents that break down the strong membrane-protein interactions.The most common type of integral proteins are transmembrane proteins, which span across the lipid bilayer such that one end contacts the cell interior and the other end touches the exterior.When proteins cross the lipid bilayer, they usually adopt an α-helical configuration ( Fig. 2) [73,78].Single-pass membrane proteins cross the membrane only once, while the multiple-pass membrane proteins are crossing the membrane several times.Many of the integral membrane proteins function as ion channels or transporters, regulating the influx of ions/molecules between the extracellular and intracellular spaces.Cell surface receptors, linkers, enzymatic proteins, and proteins responsible for cell adhesion [66] are all classes of integral membrane proteins and hormone receptors, Band3, rhodopsin, histocompatibility antigens, glycophorin, and Na + and K + channels are some examples of integral proteins in a cell membrane.Recent studies have shown that activity of these proteins depends on the lipid composition and membrane-protein interactions more so than previously thought [79,80], highlight the role of lipids in the activity of these molecules fundamental for biological information transfer [81][82][83].

Peripheral proteins
Peripheral proteins temporarily bind to the surface of the membrane with weak interactions [76,84].This means that unlike integral proteins, peripheral proteins can be easily separated from lipid bilayer by either altering the pH or the salt concentration of the cell culture medium [67] two key aspects that each structure should have.First, peripheral protein have a unique amino acid sequence which allows them to bind and congregate on the surface of the membrane [85,86].Second, there is no hydrophobic region of amino acids in peripheral proteins structure, therefore they can attach to membrane surface without being locked onto it [85,86].Classic examples of peripheral proteins include Cytochrome c, spectrin in erythrocytes, myelin basic protein, and acetylcholinesterase in electroplax membranes [87,88].The primary role of peripheral proteins is to provide a point of attachment for other components to the cell membrane.For instance, both membrane cytoskeleton and components of extracellular matrix are linked to the cell membrane through peripheral proteins, thus they help the cell to maintain its shape while the membrane remains flexible to bend based on the cellular functions [89].Besides the structural supports, peripheral proteins are involved in various other functions including cell communication, energy transduction, and molecule transfer across the membrane [89].

Glycoproteins
Glycoproteins are a class of proteins which have carbohydrate chains (in the form of oligosaccharides) covalently attach to the main protein body [90,91].The presence of carbohydrate chains can dramatically alter the intrinsic properties of a glycoprotein such as the size, charge, solubility, structure, and accessibility [92].Depending on how and where carbohydrate chains attach to the protein, the glycoproteins are classified into three categories; N-linked glycoproteins, O-linked glycoproteins, and non-enzymatic glycosylated glycoproteins [90,93].In N-linked glycoproteins, the carbohydrate chains are attached to the the nitrogen atoms of the amino acid asparagine.In O-linked glycoproteins, the carbohydrate chains are linked to the hydroxyl side chain of amino acids serine or threonine.In the third group, unlike the two others, the non-enzymatic glycosylated glycoproteins are synthesized by chemical addition of carbohydrate chains to polypeptides [73,90].In terms of functionality, glycoproteins are almost found in all living organisms and serve a number of important roles as structural molecules, immunologic molecules, transport molecules, receptors, enzymes, and hormones [90,94].For instance, the glycoproteins on the membrane surface form bonds with the extracellular matrix, helping the cell stabilize  its membrane structure.In addition to that, the glycoproteins receptors and antigens facilitate cell-cell recognition and interaction which are a key factor for immune system functionality [94].

Mechanical viewpoint
Theoretical approaches are complementary techniques that have been developed in the last few decades to how cells regulate their function through geometry, mechanics, and signaling.[11,58,[95][96][97].In general, theoretical approaches can be classified into discrete and continuum models.In discrete models, the equations describe atoms' motion with in interaction with each other, through a set of bonded and non-bonded potentials, are solved by Molecular Dynamics (MD) or Coarse-Grained (CG) simulation techniques [98,99].Tracing all atoms in a system makes this model suitable for exploring the nature of the biological process at the molecular level such as the biochemistry underlying the lipid-lipid or lipid-protein interactions, which are typically very difficult to detect experimentally.However, the high computational cost of MD or CG simulations limit the applications of discrete models just to phenomena at nanoscopic length and time scales [95,100,101].
On the other hand, there is a continuum approach that deals with the membrane as a continuous surface with average properties [95].Indeed, the small length scale of the membrane constituents (∼ 3-6 nm) compared to the length scales of biological phenomena (∼ 100 nm-µm) allows us to define the complex membrane as a single continuum surface [95].
The most popular and widely used model in continuum framework is the Helfrich model that was proposed in 1973 [102].
In this model, the membrane is considered as a thin elastic shell that can bend such that at all the times the lipids remain aligned and normal to the membrane surface.In addition, this model presumes that the curvature of the membrane is much larger than the thickness of the bilayer [102].Under these assumptions, Helfrich proposed an energy function for the membrane system that depends only on the mean and Gaussian curvatures of the membrane as [102] where W is total strain energy of the membrane due to bending, H is the membrane mean curvature, K is the membrane Gaussian curvature, and κ and κ G are the membrane properties which are called the bending and Gaussian modulii respectively.The integration in Eq. 1 is over the entire membrane surface area ω and dA is the area element.We describe the geometrical concepts of membrane curvature in Box. A.

Simulation techniques
From a mechanical perspective, cell membrane deformation can be characterized by balance laws for mass and momentum.
Simplifying these mass and momentum governing equations in continuum framework results in partial differential equations (PDEs) [103].To solve the PDEs, first step we need to define the constitutive relationship for membrane deformation such as Helfrich bending energy (Eq.1).Other forms of suggested constitutive equations including the effects of proteins are presented in Sec. 4.
Besides the need for a constitutive equation, the derived PDEs from cell mechanics are usually higher order and highly nonlinear differential equations.Therefore, in most cases, analytical solution are not possible and the equations are often solved numerically.Over the last few decades, various computational approaches have been developed to solve the set of governing PDEs including the boundary value problem for axisymmetric coordinates [21,37,44 element methods [106][107][108], Monte Carlo methods [109][110][111], finite difference methods [112,113], and the phase field representation of the surface [114][115][116].Each of these methods has its own advantage and disadvantage and depending on the complexity of the problem, one or more of them can be implemented.
A major challenge in modeling membrane protein interactions is identifying a constitutive relationship that captures the different levels of complexities associated with membrane protein interactions.In what follows, we discuss some of the popular models used for such purposes along with their applications.We then discuss where new constitutive relationships relationships are needed and how these can be experimentally parameterized.

Spontaneous curvature model
In the spontaneous curvature (SC) model, it has been suggested that the interaction between proteins and surrounding lipids changes the local membrane properties particularly the preferred -called spontaneous -curvature of the membrane [18,[117][118][119].In this case, the induced spontaneous curvature is a parameter that reflects a possible asymmetry between the two leaflets of the bilayer.This can be the result of any membrane bending mechanisms such as phase separation of membrane proteins into distinct domains, amphipathic helix or conically shaped transmembrane protein insertion, protein scaffolding, or protein crowding ( Fig. 5A).In reality, a combination of all these mechanisms can occur simultaneously and the local value of spontaneous curvature can then be interpreted as a single measure of the curvature-generating capability of the membrane-protein interaction [17,18].In a continuum framework, the most common model for induced spontaneous curvature is the modified version of Helfrich energy (Eq.1), given by [44,118,120,121] where C is the spontaneous curvature and its effective strength depends on the membrane composition, temperature, the membrane thickness, the protein density, and the membrane area coverage by proteins [102,122].
Modeling the net effect of membrane-protein interaction as an induced spontaneous curvature (Eq.2) has provided great insight into various aspects of membrane deformation, from vesiculation in caveolae and endosomal sorting complexes to cylindrical shapes of membrane endoplasmic reticulum (ER) [123][124][125].By using the SC model, recent studies have shown for example how a line tension at a lipid phase boundary could drive scission in yeast endocytosis [21,126,127], or how a snapthrough transition from open U-shaped buds to closed Ω-shaped buds in Clathrin Mediated Endocytosis (CME) is regulated by membrane tension [37,44].Furthermore, the experimentally observed change in membrane tension (spontaneous tension) in response to protein adsorption [128][129][130], can be explained in the context of the SC model [104,120,122].The SC model has also been used to elucidate the role of varying membrane tension due to spontaneous curvature papers [104,120,122].While the SC model has been very effective in capturing large scale deformations of the membrane, it doesn't take into account the protein density or the curvature induced by each protein.Let us consider the membrane as a two dimensional surface in a three dimensional Euclidean space (Fig. A3).At each point on the surface, there are two curvatures, κ 1 and κ 2 , which characterize the shape of the surface [131,132].These two curvatures are called principal curvatures and by the definition their values are the reciprocal of the radius of the osculating circle at the point (P) (κ 1 = 1/R 1 and κ 2 = 1/R 2 in Fig. A3) [131,132].The values of these curvatures can be positive or negative.The curvature is positive if the curve turns in a same direction as normal vector to the surface (n), otherwise it is negative [131,132].The average and the product of two principal curvatures give the mean (H) and the Gaussian (K) curvatures as [131,132]

Bilayer couple model
In order to go beyond an idealized single manifold description of a membrane, the Bilayer Couple model (BC) was proposed by Sheetz and Singer in 1974 [133].The basic idea in this model is that each lipid molecule has a fixed area and there is no lipid exchange between the two leaflets of the bilayer.Thus, any asymmetrical protein insertions into the inner and outer surfaces of the membrane causes an area mismatch between the two leaflets.This mismatch creates in-plane compression in one leaflet and extension in the other, resulting in membrane deformation to release the induced stress (Fig. 5B) [18,134].For a thin lipid bilayer with thickness (d), the area difference between the leaflets (∆A) can be expressed in terms of the mean curvature (H) as Here, instead of having a spontaneous curvature term in energy, a "hard" constraint on the area difference between the leaflets (Eq. 3) regulates the membrane curvature.This difference in the mechanism of curvature generation of SC and BC models distinguishes their predictions for the same membrane deformation [134].For example, in membrane budding transition due to thermal expansion, the prediction of the SC model is that the shape transformation is discontinuous, while based on the BC model, there are pear-shaped structures that appear as intermediates and the transition of shapes is continuous [134].

Area difference elasticity model
In 1980, the Area Difference Elasticity (ADE) model was developed by Svetina et.al, [135,136] to combine both SC and BC models including the missing macroscopic details of membrane bending phenomena.To better explain the physics underlying this model, we consider a flat membrane that bends downward due to different protein concentrations on two sides of the membrane (Fig. 5C).This bending, based on the single sheet descriptions of the membrane in SC model, gives rise to the spontaneous curvature term in the energy equation (Eq.2).However, if we treat each leaflet as an independent elastic plate -as was suggested in the BC model -we can then see that besides the curvature, the area of each monolayer will also change.For example, in Fig. 5C, the outer monolayer is stretched and the inner one is compressed.The energy associated with the membrane bending and this relative change in the monolayers areas is given by [134,137,138] Bending energy where κ r is called the nonlocal membrane bending modulus, and A is the total surface area of the neutral plane.∆A 0 and ∆A are the relaxed (initial) and the bent area differences between the membrane leaflets respectively (∆A 0 = A 0,out − A 0,in and ∆A = A out − A in , in which A out is the area of the outer layer and A in is the area of the inner layer).In Eq. 4, κ and κ r are both in order of K a d 2 , where K a is the area stretching modulus of the bilayer [134,138,139].This means that in any membrane deformation, both the terms, the bending and the elastic stretching energies, are comparable and must be considered.Using the ADE model, researchers for the first time could numerically simulate the shape transformations of the human red blood cell from stomatocyte to discocyte and to echinocyte [139][140][141][142]. Also, using the ADE model the experimentally observed vesicles were mapped into a theoretical phase diagram, enabling theoreticians to predict in what regions or range of parameters, the vesicles may become unstable [134,137].These predictions have been very useful for detecting unstable shapes, which is challenging to do experimentally.

Deviatoric curvature model
In the SC model, the induced spontaneous curvature was assumed to be isotropic, same in both directions (see Box. A).
However, not all proteins are rotationally symmetric and have intrinsically anisotropic structures such as banana shaped BIN-amphiphysin-Rvs (BAR) proteins (Fig. 5D) [143][144][145].These proteins can produce different curvatures in different directions and this difference is required for the formation of nonspherical structures such as membrane tubular protrusions [146,146,147].In order to take into account the anisotropic contribution of protein coats or inclusions in the continuum approach, Kralj-Iglic et.al proposed a Deviatoric Elasticity (DE) model [148].In this model, each complex protein structure is simplified as a one-dimensional curve that lies on the membrane.The orientation and the position of the proteins in the plane of the membrane are important factors since an extra term is needed in the for adjusting the actual local curvature of the membrane to the intrinsic curvatures of the proteins [148,149].The membrane-free energy that was suggested by the DE model is given as [148,150] Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 4 September 2018 doi:10.20944/preprints201809.0055.v1 Peer-reviewed version available at Biomolecules 2018, 8, 120; doi:10.3390/biom8040120 where D is the membrane curvature deviator and D 0 is the spontaneous membrane curvature deviator.Since the DE model was proposed, there have been many modeling efforts to explain how BAR proteins accumulation in membrane necks stabilize membrane tubular protrusions without any cytoskeleton supports [151][152][153][154]. Effectively, derivation of the Euler-Lagrange governing equations by a variational approach [155], provides a platform to systematically explore the impact of the induced stresses by anisotropic curvatures on the morphology of tubular structures [21].

Protein aggregation model
Aggregation of cytosolic proteins on the membrane surface or phase separation of bilayer proteins into specific domains have been observed in many biological processes [156][157][158][159].This aggregation of proteins not only creates a concentration field on the membrane surface but also results in additional contributions to the membrane energy due to compositional heterogeneity and the entropic interactions of bulk proteins with the lipid bilayer (Fig. 5E) [160][161][162].While the exact form of the free energy is still a matter of debate and has not been verified experimentally yet, a simple model based on thermodynamic arguments is given as [160,161,163], Entropic energy Energy due to protein aggregation Energy penalty due to compositional heterogeneity dA, where T is the environment temperature, a is the surface area occupied by one protein, φ is the relative density of the proteins, and J is the aggregation potential.In Eq. 6, the first term is the conventional Helfrich bending energy with induced spontaneous curvature [102].The second term represents the entropic contribution due to the thermal motion of proteins in the membrane [160,164].The third term gives the aggregation energy, and the last term describes the energetic penalty for the spatial membrane composition gradient [160,163,164].The suggested protein aggregation model mainly used for theoretical analysis of dynamic phase transitions of coupled membrane-proteins-cytoskeleton systems in membrane protrusions such as microvilli and filopodia [160,[165][166][167].This model also reveals one interesting fact that in addition to the induced deviatoric spontaneous curvature of the BAR domain proteins, the associated energy with their aggregation at membrane necks facilitates tubular structures stability.[154,168].A major open question in the field is the relationship between protein density, size, and spontaneous curvature.Although current models use a linear proportionality [104,161,169], this choice of functions is critical in determining the energy.

Protein crowding
Protein crowding is a recently discovered curvature generating mechanism that has challenged some conventional paradigms about the role of involved molecular machinery in a robust cell shape change [34][35][36][170][171][172] raises a controversial prediction that the large disordered domains of BAR domains proteins induce crowding pressure that promotes membrane fission instead of stabilizing the membrane [173].Additionally, another set of experiments reported that the induced crowding pressure by a high concentration of cargo proteins on one side of the ER works as an obstacle, which opposes membrane bending and inhibits the vesiculation by coat protein complex II (COPII) [174][175][176].
The essence of the crowding mechanism is that the lateral collisions between the membrane-bound proteins on one side of the membrane generate a steric pressure that causes the membrane to bend away from the bulk proteins (Fig. 5F) [35,177,178].As the density, the size or the mobility of the bound proteins increase, the induced steric pressure becomes larger, which results in a more significant membrane bending [35,36].Modeling the free energy associated with protein crowding is more difficult because it profoundly depends on the specific composition of the underlying membrane as well as the lateral confinement of the membrane-bound proteins [179,180].However, in a recent paper, a simple 2D hard-sphere gas model based on the Carnahan-Starling approximation has been proposed to describe the free energy of the crowding mechanism [35,181].To better visualize it, let us consider a membrane that is crowded with different protein concentration on each side as shown in Fig. 5.If we model each protein as a hard-sphere gas particle that exerts certain pressure to the membrane surface, the work that is done by this pressure to bend the membrane according to the standard thermodynamics is given by [182] where p in and p out are the induced steric pressure by the crowding proteins on the inner side and the outer side of the membrane respectively.This induced pressure (denoted by p here ) for a 2D hard-sphere gas protein can be expressed as [180,183,184] where k B is the Boltzmann constant and p R (φ) is the reduced gas pressure depending on the relative density of the protein given as [184] p R (φ Eq. 9 is known as a 2D version of the Carnahan-Starling equation.Based on this equation, at low protein density, the reduced pressure is simply proportional to φ, but as the gas density increases, the non-linear terms play larger roles and should be considered.

Hydrophobic mismatch
Transmembrane proteins embedded in the cell membrane have a hydrophobic region that is in contact with the hydrophobic region (lipid acyl chain) of the lipid bilayer.Energetically, it is then favorable that both hydrophobic regions have approximately a same thickness in order to prevent the exposure of the hydrophobic surfaces to the hydrophilic environment.
However, there are various proteins with different lengths in a single membrane [185,186].On the other hand, one protein with a same length can be surrounded by lipid bilayers with different thicknesses [187,188].This difference between thicknesses of hydrophobic regions of a transmembrane protein (d p ) and the lipid bilayer (d l ) is called hydrophobic mismatch.There are different adaptation mechanisms that either the protein or the bilayer can utilize in order to avoid mismatch, the lipid bilayer can be stretched or compressed respectively to adjust the length of hydrophobic regions [190,191].Another possibility is when the hydrophobic part of a transmembrane protein is too thick or too short as compared to the hydrophobic bilayer thickness.In this case, protein aggregation on the membrane or protein surface localization can efficiently minimize the exposed hydrophobic area [192,193].Also, for proteins that have helices that are too long compared to the thickness of the membrane, helix tilt is one possible mechanism to reduce the protein effective hydrophobic length [187,194,195].Several theoretical approaches have been developed to incorporate the energy cost and thermodynamic effects of membrane-protein interactions in term of hydrophobic mismatch [196][197][198][199].
Thus, in addition to the models described above, there are additional considerations to the energy that have been suggested by numerous studies such as higher order bending terms [132,200,201], lipid volume constraints [202], the impacts of a protein shape on membrane deformation [203], and the electrostatic energy between a membrane and proteins [204][205][206][207].

Future perspective and challenges
Although the models discussed above have provided insight into some fundamental questions about the molecular machinery of cell shape regulation, all of them have been developed based on simplifying assumptions that need to be revisited in the pursuit of closing the gap between experiment and theory.In order to achieve this goal, multidisciplinary efforts between physicists, mathematicians, engineers, and biologists are required to match different pieces of this cell biology puzzle.
Here, we highlight some current challenges that we believe must be considered in the next generation of continuum models.
• Membrane deformation is a dynamic process, surrounding fluid flow, thermal fluctuation, and diffusion of proteins actively regulate the shape of the membrane at each instance [11,169,[208][209][210][211][212].Currently, the models for membranes at mechanical equilibrium are quite well developed but the models for a dynamic process have not been as well-developed and the community must invest some effort in this aspect.
• In vivo, multiple mechanisms coupling membrane deformation and cytoskeletal remodeling are commonplace (Fig. 6A).Therefore, the models should be extended to include the dynamic effect and rearrangement of the actin cytoskeleton layer underneath of the membrane.
• Membrane deformation and protein absorption/rearrangement are often considered as two separate processes with little to no impact on each other.However, recent studies show that proteins can sense the membrane curvature (Fig. 6B).Therefore, indeed, there is a feedback loop between the protein distribution and the membrane configuration.
• Cell shape can control signal transduction at the plasma membrane, and on the other hands, intracellular signaling changes the membrane tension [220] (Fig. 6C).This coupling between the cell shape and the signaling network inside the cell should be further understood in terms of both quantitative experimental and theoretical biology.
• As discussed above, membrane deformation is a multiscale phenomena that results from the reorientation of lipids to large scale change in membrane curvature.This suggests the extension of available models toward multiscale models that could represent each biological process over multiple length scales [101,221].
Despite these challenges, with increasingly quantitative measurement techniques available experimentally, ease of access to high throughput computing systems, and interdisciplinary training the next generation of scientist leaders, the future of theoretical modeling of biological membranes and cellular membrane processes is brighter than ever.

Figure 2 .
Figure 2. Schematic depiction of the composition of a cellular membrane.There are two layers of amphipathic lipid molecules that self-assemble to form the bilayaer.In each layer, the hydrophilic head groups form the outer surface and the hydrophobic tails face toward each other in the interior region.The distribution and organization of lipids and different proteins can vary from cell to cell.The cell membrane is decorated with many different molecules including peripheral proteins, integral proteins, and carbohydrate molecules.

Figure A4 .
Figure A4.Axisymmetric coordinates with z as the axis of rotation.

Preprints(Figure 5 .
Figure 5.The mechanisms of membrane curvature generation due to protein interactions in different continuum elastic models.(A) Local protein interactions with membrane produces a spontaneous curvature filed.(B) The asymmetric insertion of conical proteins on one side of the membrane results in the expansion of the upper leaflet and compression of the lower leaflet.(C) The are of each membrane leaflet changes due to membrane bending.(D) Rotationally non-symmetric proteins generates anisotropic curvature.(E) Aggregated proteins on the membrane surface creates a spontaneous curvature filed and also have entropic interactions with the membrane.Here φ represents the relative density of the accumulated proteins.(F) The induced pressure (p) by crowding proteins drives membrane bending.a is the surface area occupied by one protein.

Figure 6 .
Figure 6.Perspective for the the future of theoretical models for membrane curvature generating mechanisms.(A) The coupling between membrane shape, membrane curvature, and membrane proteins distribution.The convex proteins (indicated with red cones) aggregate and flow toward the hill where the membrane curvature is negative (assuming the normal vector to the surface is outward).On the other hand, the concave proteins (represented by blue cones) accumulate and move toward the valley where the membrane curvature is large and positive[161].(B) Various mechanisms are involved in trafficking including amphipathic helix insertion into the bilayer, protein scaffolding, cargo-receptor crowding, forces from actin polymerization, and lipid phase separation[213,214]. (C) The coupling between the formation of a filopodial protrusion and the intracellular signaling inside the cell[215].

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 4 September 2018 doi:10.20944/preprints201809.0055.v1
. A peripheral protein can have different structures, but there are [34]170]wiak et.al.reportedthatconfining a sufficiently high concentration of his-tagged Green Fluorescent Proteins (GFP) to a local region can deform the membrane into buds or tubules in the absence of any protein insertion into the lipid bilayer[35,170].Furthermore, Snead et.al.showed that crowding among membrane-bound proteins can also drive membrane fission[34].This paper