Membrane mediated aggregation of curvature inducing nematogens and membrane tubulation
hhttp://dx.doi.org/10.1016/j.bpj.2012.12.045, Biophysical Journal
Membrane mediated aggregation of curvature inducing nematogens andmembrane tubulation
N. Ramakrishnan ∗ and P. B. Sunil Kumar † Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
John H. Ipsen ‡ MEMPHYS- Center for Biomembrane Physics, Department of Physics and Chemistry,University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark (Dated: September 30, 2018)
Abstract
The shapes of cell membranes are largely regulated by membrane associated, curvature active, proteins. We usea numerical model of the membrane with elongated membrane inclusions, recently developed by us, which possesspontaneous directional curvatures that could be different along and perpendicular to its long axis. We showthat, due to membrane mediated interactions these curvature inducing membrane nematogens can oligomerizespontaneously, even at low concentrations, and change the local shape of the membrane. We demonstrate that fora large group of such inclusions, where the two spontaneous curvatures have equal sign, the tubular conformationand sometime the sheet conformation of the membrane are the common equilibrium shapes. We elucidate thefactors necessary for the formation of these protein lattices . Furthermore, the elastic properties of the tubes, liketheir compressional stiffness and persistence length are calculated. Finally, we discuss the possible role of nematicdisclination in capping and branching of the tubular membranes.
PACS numbers: PACS-87.16.D-, Membranes, bilayers and vesicles. PACS-05.40.-a Fluctuation phenomena, random pro-cesses, noise and Brownian motion. PACS-05.70.Np Interfaces and surface thermodynamics ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ phy s i c s . b i o - ph ] F e b . INTRODUCTION Membrane shape deformations are key phenomena in a multitude of cellular processes, including pro-tein sorting, protein transport, organelle biogenesis and signaling. In the last decade a profusion ofregulatory proteins facilitating such shape changes of the cellular membranes has been unravelled, withthe BAR protein superfamily [1], the Pex11 family [2] and coat proteins [3] as notable examples. Thepossibility of such mechanisms has long been anticipated in the biophysical literature[4, 5]. Howeverthe experimental and theoretical difficulties involved have hampered the establishment of a quantitativebasis for interpreting such phenomena in cell biology. Recently, we had overcome one such obstacle bythe establishment of a computer simulation technique to study how the cooperative effects of membraneinclusions, imposing a curvature along the direction of its orientation, remodels vesicular membranes[6].In this work we aim at describing, from a theoretical point of view, the effect of a large group of thesemembrane curving proteins, which can be considered as effectively elongated objects in the plane of themembrane. We consider inclusions with approximate π -symmetry, i.e. the protein can be consideredas essentially indistinguishable from its form rotated by 180 ◦ around the protein center in the planeof the membrane. The membrane inclusions we consider, has thus some similarity with nematogensin 3-D nematic liquid crystals. However, they are embedded in a membrane and may couple to itsgeometry, and it is only the part of the protein in contact with the membrane, which will be subjectto these symmetry requirement. Therefore, we cannot consider these membrane inclusions as simpleliquid crystal nematogens restricted to Euclidean two dimensional surfaces. In the following we will referto such membrane inclusions as membrane nematogens. Large groups of membrane curving proteinsfall into this category of membrane nematogens. An example is the BAR proteins( proteins containingboth BAR domains and/or N-terminal helices ), where both the N-terminal amphipatic helices and thebanana-shaped, positively charged, dimeric interface with the membrane, induces directional curvature[7–12]. The caveolin protein family[13], which form dimers and are bound to the membrane by a pairof hairpins and the reticulon, DP1 and Yop1p involved in the formation of smooth ER [15, 16], andare anchored to the membrane by two similar hairpins are also examples. The cell biology literaturehas provided good evidence for that the insertion of amphipathic helical peptide sequences not onlyprovide a binding mechanism, but also gives rise to local modulation of the membrane curvature [17, 18].More solid, quantitative support for this conjecture is given from biophysical experiments [19] and theory[20] . Furthermore, biophysical studies has demonstrated that curvature active membrane inclusionshave dramatic effects on the cooperative behavior with a complex interplay between lateral ordering andmembrane shape. However, the detailed mechanisms leading to the specific complex membrane-proteinstructures have not been characterized. This work will elucidate some aspects of these mechanisms forthe membrane nematogens.Some of the key processes involved in the structural organization of membrane nematogens described inthe cell biology literature, can be categorized as follows: (1) the aggregation of the nematogens - the pro-cess where membrane proteins upon activation and/or binding to the membrane spontaneously aggregateand form functional cluster of proteins in the membrane[9–11, 14], (2) Tubulation of membranes, wherethe aggregate and the membrane develop tube-like membrane structures (e.g. sorting endosomes[21, 22]and Mitochondrial outer membrane[23], formation of T-tubules in Drosophila [24]) and (3) The formationof protein lattices , often characterized by helical arrangement of the proteins spiraling around the tubularmembrane, e.g. for dynamin [25, 26] or caveolin [27].In this work, we will demonstrate by numerical analysis of a possible physical model, which capturesthe membrane conformations and the organization of in-plane nematogens, that the above mentionedprocesses directly results from the cooperative thermodynamic behavior of the nematogens coupled tothe flexible membrane. Also, we will discuss aspects of the stability of membrane tubes and the formationof the edges for membrane sheets. Our model gives a coarse description of the membrane, which captureproperties of the membrane which are essential for its large scale organization. Despite the simplicity ofthe model, the parameter space is too large for a comprehensive discussion of it’s phase behavior. Rather,we will focus on some generic features of the model which may well give a framework for interpreting theexperimental observations of cellular membrane morphogenesis. Previously, protein induced membranetube formation has been considered by a phenomenological model involving scalar fields [28], and thecoupling between membranes and inclusions with directional curvature was modelled in [29–32].The paper is organized as follows: In Section II the physical model of the interacting system of mem-brane and membrane nematogens are presented, while details about the numerical analysis is given in
Supplementary Materials . Section III on
Results and Discusion present some generic properties of2he model and discuss their possible relevance to experimental results. In section III A-III C the aggrega-tion of proteins and membrane domain formation, membrane tubulation and formation of protein lattices are described in the framework of the model. Section III D discusses the elastic properties of membranetubes and their relevance to observable effects. Much of the characterization of the elasticity of proteinlattices is based on a continuum version of the model discussed
Supplementary Materials . SectionIII E discusses mechanisms of closing, capping and branching of membrane tubules and the possible roleof nematic point defects. Section III F describes aspects of the stability of membrane tubules with moremembrane curvature components. In section III G the interplay between sheet and tubule formation isdescribed and possible implications for cell organel morphology is discussed. Some perspective on themodeling of membrane morphogenesis is given in
Conclusion , Section IV. We will in this work specializeto properties of membranes with inclusions which posses directional curvatures of equal sign. We willconsider cases with different signs of the directional curvatures in a seperate publication.
II. MODEL
The modeling of the effects of in-plane nematogens on membrane structure, will in this work, betreated with a discretized description of the surface as a randomly triangulated mesh. A continuoussurface conformation is approximated by a collection of triangles glued together to form a closed surfaceof well defined topology. A triangulated surface, with spherical topology, thus consist of N verticesconnected by N L = 3( N −
2) links, which enclose N T = 2( N −
2) triangles. Each vertex v is assigneda position (cid:126)X v . This tesselatations of the surface form the basis for a coarse grained description of themembrane, where only the gross features of the structure and interactions are important. planetangent v ˆ N ( v ) c v c v ˆ t v ˆ t v v ˆ n ( v ) ϕ v ˆ t v ˆ t v (a) (b) (c) Mixed phase demixing and constriction
Activation of field B
FIG. 1: (a) A one ring triangulated patch around a vertex v . The shaded region represents the tangent planeat v and ˆ N ( v ) its corresponding normal. c v and c v are the maximum and minimum principal curvatures,respectively, along principal directions ˆ t v and ˆ t v . (b) Illustration of the nematic field vector ˆ n defined on thetangent plane of vertex v . (c) A vesicle of spherical topology with spatially random surface nematogens. The triangulation and the vertex position form together a discretized surface, a patch of which is givenin fig: ?? . The geometry of the continuous surface, which is approximated by the triangulated surface,can now be characterized by a number of surface quantifiers, e.g. the curvature tensor, the principaldirections (ˆ t v & ˆ t v ), the corresponding principal curvatures ( c v & c v ) and surface normal, ˆ N ( v ), ateach vertex v . The details can be found in [6]. The discretized Helfrich’s free energy[34] can then beevaluated as H c = κ N (cid:88) v =1 A v H v (1)where H v = ( c v + c v ) / v and A v is the area of the surface patch occupiedby the triangles adjacent to vertex v . κ is the bending rigidity of the membrane. Furthermore we are ina position to calculate the directional curvatures along and perpendicular to a unit vector ˆ n along thesurface by use of Gauss formula: H v, (cid:107) = c v cos ϕ v + c v sin ϕ v ,H v, ⊥ = c v sin ϕ v + c v cos ϕ v , (2)3here ϕ v is the angle between ˆ n and the principal direction ˆ t v . Such an orientational spontaneouscurvature may be induced by a membrane nematogen with an orientation in the plane of the membranegiven by ˆ n . In addition to the interaction with the membrane, nematogens may tend to orient alongeach other at close proximity due to steric, electrostatic and dispersion interactions [35]. In the presentstudy, we focus only on the two dimensional orientational interactions promoted by the underlying, non-planar, fluctuating membrane [36–40]. The π -symmetry of the individual nematogens dictates that thesimplest form of their self interaction should be of the type cos ( θ vu ) and sin ( θ vu ), where θ vu is the anglebetween ˆ n v and ˆ n u at neighboring vertices. We choose to represent the interactions between membranenematogens by an extension of the well-established Lebwohl-Lasher model of nematic ordering in presenceof vacancies, here implemented on a triangulated surface model of a membrane.The nearest neighbor interaction between the nematogens is composed of an isotropic componentrepresented by an interaction strength J and an anistropic (quadrupolar) correction measured by theinteraction constant (cid:15) LL . The total interaction between the membrane nematogens thus takes the form H field = (cid:88) (cid:104) vu (cid:105) (cid:26) − J − (cid:15) LL (cid:18)
32 cos ( θ vu ) − (cid:19)(cid:27) I v I u , (3)where the sum is over nearest neighbour vertices. I v = 0 , v ” is occupied by a nematogen and zero if otherwise. The calculation of the θ vu is non-trivial,since the angle between spatially separated nematogens are measured after the parallel transport of vec-tors along the curved surface [6]. With this measure of the angular differences, Eq.(3) models the in-planeinteractions of the nematogens mediated by the membrane. The direct distance dependent interactionsthrough the cytosol is not taken into account in this model of membrane-protein conformations. Suf-ficiently large, positive (cid:15) LL favors in-plane ordering of the nematogens. The effect on the anisotropicelasticity of the membrane due to the nematogens, in analogy with the discretized Helfrich free energy,takes the form [6]: H nc = N (cid:88) v =1 (cid:26) κ (cid:107) (cid:16) H v, (cid:107) − H (cid:107) (cid:17) + κ ⊥ (cid:0) H v, ⊥ − H ⊥ (cid:1) (cid:27) I v A v (4) H (cid:107) and H ⊥ are the spontaneous curvatures along ˆ n and ˆ n ⊥ , while κ (cid:107) and κ ⊥ are the correspondingdirectional membrane bending elastic constants.Self avoidance of the discretized surface is ensured by imposing constraints on the neighboring vertexdistance and on the dihedral angles between neighboring faces[6]. The equilibrium properties of thediscretized surface can now be evaluated by standard Monte Carlo sampling of Boltzmann’s probabilitydistribution ∼ exp (cid:16) − k B T [ H c + H field + H nc ] (cid:17) at fixed concentrations of the membrane nematogens.A general description of the implemetation of such numerical models and further details about thesimulations are given in [6].Finally, we will make some considerations about length scales. The lattice model is a highly coarse-grained representation of the membrane, designed to capture the large length-scale properties of mem-branes with inclusions. Therefore, the triangulated surface represent a collection of membrane patcheswith a characteristic length scale. A natural choice of length scale is to identify a tether length with thelateral extension of a membrane inclusion. Some examples here are CIP4 F-BAR with a length of 22nm[41] or dynamin which extent about 25 nm [25].The computer simulations of the discrete model provide us with insight into the nature of equilibriumconfigurations for a choice of model parameters. To complement the numerical simulations, it is usefulto consider the corresponding continuum model in the limit of membrane nematogen with 100% surfacecoverage. It is an extension of Helfrich’s bending free energy functional [42, 43] F = (cid:73) dA (cid:26) K A ∇ ˆ n : ∇ ˆ n ) + κ H ) + κ (cid:107) H n, (cid:107) − H (cid:107) ) + κ ⊥ H n, ⊥ − H ⊥ ) (cid:111) where K A = √ (cid:15) LL . In Supplementary Material is presented such an analysis of the mechanicalproperties of a tubular membrane with a protein coat and analytical expressions reflecting, tube radius,persistence length and protein organization are also derived.4
II. RESULTS AND DISCUSSION
In this section we will present some key aspects resulting from the coupling of membrane nematogenproteins to lipid membranes. It will both contain results from computer simulations of the aforementionedmodel, which are non-perturbative, along with theoretical analysis of the continuum model, of a moreperturbative character to qualify the numerical finding. Throughout the discussion the parameter (cid:15) LL has a relatively high value (several k B T in a range where nematic ordering is favored). Furthermore, theimplications of our results on the experimental systems in vivo and in vitro will be discussed. A. Aggregation and membrane domain formation of membrane nematogens
A common feature of membrane nematogens is their strong tendency to self-associate, driven by theflexible geometry of the membrane - in this manuscript, we call this self associated structure to be anaggregate or a domain. Self association has been observed for a wide range of model parameters κ (cid:107) , κ ⊥ , H (cid:107) and H ⊥ . All results presented in the following corresponds to system size with N = 2030vertices. When the fraction of nematogens φ A = 0 . (cid:15) LL = 3 and J = 0 (in units of k B T ), the flexiblemembrane with curvature coupled to the nematic orientation, gives rise to co-existence of nematicallyordered domains and the isotropic dilute phase, this is shown in figure-2(b). This is to be compared withthe planar Lebwohl-Lasher model on a random triangular lattice, at the same concentration, where theisotropic phase is stable ( see Supplementary Materials ). Additional direct repulsive interactions J ≤ − . J promotes the aggregation and can change the aggregate shape as shown in Fig. (2)(c). (c) Tubular(a) Spherical (b) Disc − < J < − . − . < J < . . < J < FIG. 2: Equilibrium membrane conformations, with φ A = 0 . κ = 20, κ (cid:107) = 5 and H (cid:107) = 0 . (cid:15) LL = 3, fordifferent range of J . The effect of concentration is shown in Fig.(3) for surface coverage in the range φ A = 0 . − .
7, whichdisplay a series of complex shape deformations connected to different aggregate structures. More detailswill be discussed in section III G.The aggregation of membrane nematogens also has a temporal aspect. In Fig.(4) we have shown aMonte Carlo time series, for a membrane coverage of 10% nematogens, to illustrate the sequence ofdomain formation and membrane curvature induced changes leading to the equilibrium structure. Themembrane nematogens in an initial randomly dispersed orientation assemble into smaller orientationallyordered domains mediating the final equilibrium structure. These ordered domains often appear asmetastable configurations, which either disperse again due to lateral fluctuations or they will eventuallyfunnel into a equilibrium domain configuration. The Monte Carlo dynamics does not reflect the physicalkinetics very well, but is useful in identifying kinetic paths connecting various metastable states that leadto the global minima [44, 45] .The aggregation of membrane inclusions mediated by membrane curvature deformations and fluctuationsis not specific for nematogens, but is a more general phenomena for membrane curvature active com-ponents. It has been well understood in the framework of models for curvature instabilities [5, 19, 46],5 a) φ A = 0 . φ A = 0 . φ A = 0 . φ A = 0 . φ A = 0 . φ A = 0 . φ A = 0 . FIG. 3: Equilibrium configurations for varying composition. κ = 10, κ (cid:107) = 5, κ ⊥ = 0, H (cid:107) = 0 . H ⊥ = 0, J = 0, φ A = 0 . − . (cid:15) LL = 3. (a) T MCS = 0 (b) T MCS = 1 . × (c) T MCS = 17 . × FIG. 4: Aggregation of membrane inclusions for κ = 10, κ (cid:107) = 5, κ ⊥ = 0, H (cid:107) = 0 . H ⊥ = 0, J = 0, φ A = 0 . (cid:15) LL = 3. Monte Carlo time series showing a) random initial configuration of membrane nematogens, b)intermediate state with multiple nematic domains and c) equilibrium conformation where all the small domainscoarsen into a single patch. and has also been demonstrated that simple amphiphathic inclusions, e.g. antimicrobial peptides likeMagainin or Melittin [19, 47] and viral membrane active proteins like NSB4 of Hepetites C[48].The self-association of these membrane components thus needs not to be facilitated by strong directattractive interaction amongst them. The coupling to the membrane geometry provides additional indirectmembrane conformation mediated attractive forces making them to slip into bound structures involvingboth the proteins and membrane curvature. However, the structure of the aggregates are dependent onthe details of the molecular structure and the direct interactions. A general feature of these aggregates isthat they appear as nematically ordered domains, where the nematogens form elongated oriented patcheswith well defined curvature characteristics, e.g. ridges or cylindrical rims. In the following we will inparticular focus on the tube-like structures. 6 . Tube formation The most prevalent equilibrium domain structure is the nematic tube, where the membrane protrudeinto a cylinder like configuration with the membrane nematogens forming a coat around the cylinder.Also for tube formation the overall interaction strength ( J ) between the membrane nematogens plays asecondary role. It’s most pronounced effect is to widen the concentration range for tubulation and toenhance the line tension at the domain boundary, which can induce fission of tubes by narrowing the tubeat the boundary of the domain, as in Fig(2(c)). The effect of concentration of membrane nematogens onthe membrane tubulation phenomena is shown in Fig.(3(g)). For large concentrations of nematogens orincreasing values of J the tubes are the characteristic equilibrium structures shown in Figs. (2,3).The radius of the equilibrium membrane tubes appears to be relatively well-defined. The radius of thetube with nematic order can be calculated on basis of the continuum model Eq.(5) for the chosen modelparameters (see Supplementary Material ):¯ r = | H (cid:107) | (cid:113) κ (cid:107) + κκ (cid:107) for κ ⊥ = 0 | H ⊥ | (cid:113) κ ⊥ + κκ ⊥ for κ (cid:107) = 0 (5)So, the radius ¯ r is set by the curvature elastic model parameters, involving the absolute value of thedirectional spontanous curvatures, modulated by the curvature elastic constants. It follows from Eq.(5)that the actual tube radius is somewhat larger than the inverse directional spontaneous curvatures anddependent on the relative strength of the elastic constants.In experimental systems the membrane tube dimensions can vary considerably with different typesof proteins in the cell[1]. Membrane tubes formed in vitro by curvature active proteins also display aconsiderable variability in size. Frost et al. [41] have studied the effect of a number of mutants of CIP4F-BAR on liposomes. By mutations they find a big variations of tube diameters in the range of 50 to100 nm.Membrane tubes induced by membrane inclusions are common phenomena in biological cells, bothas more static structures like T-tubules of the muscle cells[24] or more temporal structures like sortingendosomes [49]. The examples shown in Fig.(2,3) corresponds to the cases where spontaneous curvaturesare positive, like that induced when F- BAR-domain proteins bind to organelle membranes. However, ifthe proteins induce negative spontanous curvatures, as in I-BAR domain proteins, it gives rise to tubularinvaginations as shown in Fig.(5) .As can be seen from Fig.(5), for proteins with large negative spontaneous directional curvatures, at lowconcentration ( φ A = 0 . − . φ A increases,tubes disappear and saddle like regions appear. The inner tubes and saddle like regions coexist again forlarge concentrations φ A > . C. Protein lattices
Membrane nematogens organize as nematically ordered domains and coat around the membrane toform tubes. Nematogens orient perpendicular to the tube axis when κ ⊥ = 0 , κ (cid:107) (cid:54) = 0 and H (cid:107) > κ (cid:107) = 0, κ ⊥ (cid:54) = 0 and H ⊥ (cid:54) = 0 leads to an arrangement of the nematogens along the tubedirection. For the common membrane nematogen both these parameters are non-vanishing. Such a caseis shown in Fig.(6).The helical arrangement of the membrane nematogens at the tube surface can be easily understoodconsidering that in general such arrangement will give rise to a global nematic ordering of the membranenematogens (generalized spirals are the only geodesic curves on long tubes) and the radius is set by theelastic terms. The coupled expressions for the mean values for tube radius ¯ r and the angle ¯ ϕ between thetube direction and the nematogen orientation, for different regimes of the dimension less parameter ˜ ψ :7 a) φ A = 0 . φ A = 0 . φ A = 0 . φ A = 0 . φ A = 0 . φ A = 0 . φ A = 0 . FIG. 5: Equilibrium configurations for vesicle with negative spontanous curvatures. κ = 10, κ (cid:107) = 5, κ ⊥ = 0, H (cid:107) = − . H ⊥ = 0 . J = 0 and φ A = 0 . − . ¯ r = (cid:115) κ ⊥ + κκ (cid:107) ( H (cid:107) ) + κ ⊥ ( H ⊥ ) for ˜ ψ ≤ , (cid:118)(cid:117)(cid:117)(cid:116) (cid:0) κ (cid:1) (cid:0) κ (cid:107) + κ ⊥ (cid:1) + κ ⊥ κ (cid:107) κ ⊥ κ (cid:107) (cid:16) H (cid:107) + H ⊥ (cid:17) < ˜ ψ < (cid:115) κ (cid:107) + κκ (cid:107) ( H (cid:107) ) + κ ⊥ ( H ⊥ ) ˜ ψ ≥ ψ is given by the model parameters as:˜ ψ = κ (cid:107) H (cid:107) − κ ⊥ H ⊥ | H (cid:107) + H ⊥ | ( κ (cid:107) + κ ⊥ ) (cid:115) κ (cid:18) κ (cid:107) + 1 κ ⊥ (cid:19) + κ ⊥ κ (cid:107) (7)similarly for the angle ¯ ϕ : cos ( ¯ ϕ ) = ψ ≤ , (cid:16) κ (cid:107) H (cid:107) − κ ⊥ H ⊥ (cid:17) ¯ r + κ ⊥ κ ⊥ + κ (cid:107) < ˜ ψ <
11 ˜ ψ ≥ ϕ and ¯ r are set by the model parameters. A derivation of the above expressions aregiven in Supplementary Materials . 8 a)(b)(c)
FIG. 6:
Protein lattices . Modes of a tubular membrane at different state points, for κ = 10 and (cid:15) LL = 3. ( a)Tubular conformation with (cid:104) ϕ (cid:105) = 0 for κ (cid:107) = 5, κ ⊥ = 0, H (cid:107) = 0 . ( b) Spiral modes of the tube with (cid:104) ϕ (cid:105) = 0seen for κ (cid:107) = 5, κ ⊥ = 0, H (cid:107) = 0 .
4, and ( c) rearrangement of nematics into spiral modes ( (cid:104) ϕ (cid:105) (cid:54) = 0) when κ (cid:107) = 5, κ ⊥ = 5, H (cid:107) = 0 . H ⊥ = 0 . The spiral organization of the membrane coating proteins has now been observed for many tubularmembrabe systems in vivo and in vitro , e.g for the F-BAR proteins[26, 41]. EM-tomographs of tubules ofCIP4 F-BAR on liposomes [41] show a fairly dense packing arrangement in the helical tube. The averagetube diameter is around 68 nm and the helical angle is about ϕ = 40 ◦ . Such arrangements are termed protein lattices in the cell biology literature. It is found that the helical angle ϕ of the protein lattice withrespect to the tube direction adjusts to the tube diameter such that the directional curvature is about thesame. For a similar type of experiment with dynamin[50] the membrane tubes of radius r (cid:39) ϕ = 80 ◦ ).Our simulation results suggests that the spiral organization of the protein coat on the tube need notbe a result of polymerization as often referred in the literature, but can be a self-assembly process ofthe curvature active proteins mediated by the membrane. Furthermore, the modelling suggest that these protein lattices are not conventional two dimensional lattice structures like polymerized membranes orgraphene, but rather two dimensional nematic liquid crystalline structures. In the model there is no termswhich can distinguish between a right or left turning helix, i.e. the helical arrangement is the result ofa spontaneous symmetry breaking. However, the smallest chiral symmetry breaking contribution to thefree energy can favor one of the helical orientations without having an effect on any other parameters. D. Thermal stability of membrane tubes
While our model parameter determine the mean physical properties of the tubes , e.g. the radius, weexpect the tubular membranes to display an elastic response to deformations in its shape and organizationof the membrane nematogens. This can, for e.g., be reflected in the variation of the shape characteristicsdue to thermal fluctuations. For analysis of such deformations the continuum description of coated9embrane tubes are suitable and the details can be found in
Supplementary Material . It is shownthat in general the deviations in the orientation of the membrane nematogen and the tube radius arestrongly correlated. The thermally induced fluctuations in the radius is found to be (cid:104) ( δr ) (cid:105) ¯ r = k B T π κ (cid:107) + κ ⊥ κ (cid:107) κ ⊥ + ( κ (cid:107) + κ ⊥ ) κ for 0 < ¯ ψ < , (9)where ¯ r and ¯ ϕ are respectively the equilibrium tube radius and nematic orientations and ¯ ψ = cos ¯ ϕ .We note that the relative variance in r has an upper limit k B T πκ . With a typical range κ ∼ − B Tthis ratio in Eq.(9) is of the order 0.01. For CIP4 F-BAR, reconstituted on liposomes, cryo-tomographymeasurements give ¯ r = 33nm and (cid:104) ( δr ) (cid:105) / ¯ r (cid:39) .
01 [41]. If this observed variation in tube thicknessis interpreted as frozen in thermal variations, it is in agreement with the the above theory. For rigidmembranes with large κ and/or large κ (cid:107) , κ ⊥ values we can consider the thermaly excited variations in r as small. Similarly, we can estimate the thermal fluctuations around cos ( ¯ ϕ ) for such a segment as, (cid:104) ( δ cos ( ϕ )) (cid:105) = k B T π κ (cid:107) ψ + κ (cid:107) (1 − ψ ) + κ κ (cid:107) κ ⊥ + ( κ (cid:107) + κ ⊥ ) κ for 0 < ¯ ψ < . (10)To our knowledge no experimental reports on the random variations in the helical angle has been given.A third type of deformation to consider is the bending of the tubes. It is shown in SupplementaryMaterials that when r is a constant along the tube, the free energy expression is relatively simple. Inparticular we find that the free energy of bending for a tubular membrane takes the approximate form,∆ F tot ≈ k B T l P (cid:90) L dsλ ( s ) , (11)where s is the the arc length and λ ( s ) is the line curvature along the tube, while l P is the persistencelength of the tube: l P = π ¯ r (cid:0) K A + κ + κ (cid:107) (1 − ¯ ψ ) + κ ⊥ ¯ ψ (cid:1) k B T . (12)There are few experimental measurements of the persistence length of membrane tubes with proteinlattices . For the F-BAR FBP17 producing tubes of radius r ( F BP
17) = 34nm the persistence length wasmeasured to l P ( F BP
17) = 142 µ m [41] while for amphiphysin r (amph) ∼ l P (amph) = 9 µ mwhile for dynamin r (dynamin) (cid:39) l P (dynamin) = 37 µ m. A calculations of l P from Eq.(12)solely based on κ gives predictions which are an order of magnitude too small, which indicates that otherelastic constants κ (cid:107) , κ ⊥ and K A gives the main contributions to l P . E. Capping the tubes, Defects
The formation of membrane tubes with helical coats seems to be generic for systems with membraneswith membrane nematogens. Either the helical coat has to terminate resulting in an interfacial curveseparating the coated and uncoated regions or the vesicle should sprout tubes and buds with the tipshaving a pair of point defects. The way this takes place in the tube end or at a domain boundary ismainly determined by the competition between interfacial tension, which in our model is largely regulatedby the parameter J , and bending modulus. In Fig(7) is shown that when the interaction parameter J is increased the interfacial line shrinks, first transforming the vesicle from a disk to a structure withpartially coated tubes and buds, but still no defects. Further increase in the line tension will result intubes and buds that are fully coated but minimizing the length of the interfacial line between coated anduncoated regions. It does so by either moving the interfaces to the end of the tube forming a pair ofpoint defects or deforming the membrane to form a narrow neck. Note that the line does not shrink to asingle point defect of strength +1 but instead forms a pair of +1 / π symmetry of the membrane nematogen and the strong coupling between membranecurvature and nematic orientation [52–54]. To our knowledge no details about the capping of the coatedmembrane tubes have been provided by experiments.10 op view Defectless tip side view pair of +1/2 defects (a)
J=0 (b)
J=3 (c)
J=5
FIG. 7: A partly decorated membrane with κ = 20, κ (cid:107) = 5, (cid:15) LL = 3. Shown are: (a) a disc without a defect for φ A = 0 . H (cid:107) = 0 . J = 0. (b) Tubes without defects at J = 3. ( the bottom panel shows an enlarged sideview of the tip of a tube without defects ). (c) Tubes and buds with defects when J = 5. ( the bottom panelshows an enlarged top view of the tip of a bud with defects ) F. Curvature differences leads to segregation planetangent v ˆ N ( v ) c v c v ˆ t v ˆ t v v ˆ n ( v ) ϕ v ˆ t v ˆ t v (a) (b) (c) Mixed phase demixing and constriction
Activation of field B (a) (b)
FIG. 8: (a) Tubular membrane of uniform cross section with fields A and B in the mixed phase for H (cid:107) ,A , H (cid:107) ,B =0 . , .
5. (b) Activation of field B takes H (cid:107) ,B from 0.5 to 1.25. The difference in the spontaneous curvatures leadsto phase segregation and results in the constriction of tubes. Field concentrations are respectively φ A = 0 . φ B = 0 . κ = 10 k B T , κ A (cid:107) = κ B (cid:107) = 5 k B T and (cid:15) AA = (cid:15) BB = 3 k B T . Another example of the curvature driven aggregation is demonstrated in Fig. 8. Shown in Fig. 8(a)is a tubular membrane of uniform cross section, fully decorated by two different types of membranenematogens, labelled A and B. The tube is stable, in the mixed state, when the directional spontaneouscurvature of the in-plane fields, H (cid:107) ,A = H (cid:107) ,B = 0 .
5, are the same. If a source of activation increases thespontaneous directional curvature of B to H (cid:107) ,B = 1 .
25, analogous to activation of dynamin proteinsby hydrolysis of GTP, the fields demix. The regions containing field B constrict the tube further. Theequilibrium shape of the activated membrane is observed to have successive tubular regions of largeand small curvatures ( see Fig. 8(b)), similar to the striated patterns of dynamin tubes obtained ontreatment with GTP γ s [55, 56]. For dynamin the molecular conformation and membrane tube diameteris GTP dependent [56, 57]. Furthermore it is observed that the tube constriction involves a tube twisting,suggesting a change in the helical angle ϕ [58]. In in vivo experiments, it has been demonstrated thatstructurally similar F-BAR proteins can co-localize into the same membrane tubes [14] while differingBAR proteins, like F-BAR and N-BAR, seggregate into separate membrane tubes with their respectivecharacteristic r and ϕ [41, 59]. Our analysis suggests that this recruitement of differing BAR-proteinsinto separate domains is possibly driven by their directional spontaneous curvatures.11 . Sheets versus tubes The effect of concentration, shown in Fig.(3), for surface coverages in the range 10% and 70% displaya series of complex shape deformations connected to different aggregate structures. The regime, whereinclusions stay separate, for the model parameters chosen here, appear at very low concentrations. Thefigure illustrates that for a system where the direct interactions parameter J between the membranenematogens are weak the oligomers tend form larger rim-like formations, which stabilizes disc like struc-tures of vesicles. The rims form the edges of the discs. As the concentration is increased part of the edgeturn tubular.So for a range of concentrations the disc and the tubes coexist. The tubules get more pronounced andthe discs diminishes with the increase in concentration of membrane nematogens. Recent experimentson the formation of tubular (or smooth) ER suggest that some membrane curvature active proteins,reticulon protein and DP1[15], are highly enriched in the tubular ER [60] and the ER sheet edges[16].Our results are thus in line with the idea that the concentration of these membrane nematogens are amajor determinant for the amount of ER in sheets or tubules[16]. IV. CONCLUSION
We have described the membrane curvature modifying properties of anisotropic protein inclusions,like the BAR proteins, in terms of an in-plane nematic field. We have shown that the flexibility ofthe membrane can promote aggregation and lateral domain formation of these membrane nematogens,even in the absence of self interactions. These domains can facilitate shape changes of the membrane.The equilibrium shapes obtained are strikingly similar to that seen in experiments involving curvaturemodifying proteins. Prominent structures seen are tubes and discs and coexistence of them. Dependingon the preferred curvature of the nematogens, a protein lattice with helical nematic orientation around thetube is seen. The properties of this liquid crystalline structure was further analyzed from a continuumversion of the model and the dependence of the tube radius and the orientation of the nematic withrespect to the tube axis was calculated. We also estimate the thermally induced fluctuation in thesequantities and show that they are comparable to what is seen in experiments. In addition we calculatethe persistence length of the nematogen induced tubes and show that it is in the range of experimentallyobtained values. This analysis provides the necessary basis to obtain estimates of model parameters fromexperiments on coated membrane tubes. At present the available experimental data are very limited.The present modeling provides additional support to the growing notion of the importance of localcurvature modulating proteins in membrane shape generation in biological cells. Compared to previousmodeling of the role of membrane proteins inducing directional membrane curvature we have taken intoaccount that membranes are not fully decorated, the in-plane interactions between nematogens and thearbitrary membrane shapes with spherical topology. The current work focusses only on the membraneinteracting part of the protein. Electric charges in BAR-protein are mostly localized to its membranefacing domain which in turn interact with anionic lipids and enables them to bind strongly to themembrane. One natural extension of this model is include the electrostatic interactions through cytosolbetween proteins moieties protruding out of the membrane.We emphasize that the main aim of this work is to show that anisotropic curvature induced by theinclusions can lead to aggregation and interesting shape changes. This is in contrast to the prevailingassumption that explicit protein interactions are essential for aggregation and formation of protein lattices.Though a quantitative comparison between the predictions of this model and experiments is not so easy,the model does demonstrate the possibility of generating many biologically relevant shapes of the vesicleby membrane mediated interactions alone. 12 . ACKNOWLEDGEMENTS
The computational work is carried out at HPC facility at IIT-Madras. [1] Frost A., V.M. Unger, and P. De Camilli, 2009, The BAR Domain Superfamily: Membrane-Molding Macro-molecules.
Cell
The EMBO Jour.
30, 5.[3] Antonny B, 2006, Membrane deformations by protein coats.
Curr. Opinion in Cell Biol.
18, 386.[4] Sackmann E., R. Kotulla, and F-J. Heiszler, 1984, On the role of lipid-bilayer elasticity for the lipid-proteininteraction and the indirect protein-protein coupling.
Canadian Jour. of Biochem. and Cell Biol.
62, 778[5] Leibler S., 1986, Curvature instability in membranes.
J. Phys. France
47, 507-516.[6] Ramakrishnan N, P.B. Sunil Kumar, J.H. Ipsen, 2010, Monte Carlo simulations of fluid vesicles with in-planeorientational ordering.
Phys. Rev. E
81, 041922.[7] Peter B.J., H.M. Kent, I.G. Mills,Y. Vallis, P.J.G. Butler, P.R. Evans and H.T. McMahon, 2004, BARDomains as Sensors of Membrane Curvature: The Amphiphysin BAR Structure.
Science
J. Mol. Biol.
Nature
Proc. Natl. Acad. Sci.
Biophys. J.
95, 2806-2821,[12] Shnyrova A.V., V. A. Frolov and J. Zimmerberg, 2009, Domain-Driven Morphogenesis of Cellular Membranes.
Current Biology
19, R772-R780.[13] Williams T.M. and M.P. Lisanti, 2004, The caveolin proteins.
Genome Biology
Cell
Cell
Cell
J. Cell Biol.
The EMBO Journal
25, 2898-2910.[19] Bouvrais H., K. J. Jensen, J. Brask, P. M´el´eard, T. Pott and J.H. Ipsen, 2008, Softening of POPC membranesby Magainin.
Biophysical Chemistry
Biophys. J.
95, 2325-2339[21] Kurten R.C., A.D. Eddington , P. Chowdhury, R.D. Smith, A.D. Davidson and B.B. Shank, 2001, Self-assembly and binding of a sorting nexin to sorting endosomes.
J. Cell Science
J. Cell. Biol.
J. Cell. Biol.
Genes & Dev.
15, 2967-2979[25] Sweitzer S.M. and J.E. Hinshaw, 1998, Dynamins Undergoes a GTP-Dependent Conformational ChangeCausing Vesiculation.
Cell
98, 1021-1029.[26] Low H.H., C. Sachse, A.A. Amos, and J. L¨owe, 2009, Structure of a Bacterial Dynamin-like Protein LipidTube Provides a Mechanism For Assembly and Membrane Curving.
Cell
Traffic
11, 138-150.[28] Sens P. and M. S. Turnery, 2004, Theoretical Model for the Formation of Caveolae and Similar Membrane nvaginations. Biophys. J.
86, 2049-2057.[29] Fournier J. -B., 1996, Nontopological saddle-splay and curvature instabilities from anisotropic membraneinclusions.
Phys. Rev. Lett. , 76(23), 4436.[30] Fournier J. -B. and P. Galatola, 1998, Bilayer Membranes with 2D-Nematic Order of the Surfactant PolarHeads.
Braz. Jour. of Phys. , 28, 329.[31] Kraij-Iglic V., V. Heinrich, S. Svetina, S. and B. Zeks, 1999, Free energy of closed membrane with anisotropicinclusions.
Eur. Phys. J.
B 10, 5-8.[32] Dommersnes P. G. and J.-B. Fournier, 1999, N-body study of anisotropic membrane inclusions: Membranemediated interactions and ordered aggregation
Eur. Phys. J.
B 12, 9-12.[33] Iglic A., T. Slivnik, and V. Kralj-Iglic, 2007, Elastic properties of biological membranes influenced by attachedproteins.
J. Biomech.
40, 2492-2500.[34] Helfrich W., 1973, Elastic properties of lipid bilayers: theory and possible experiments.
Naturforsch Z.
Annals of the New YorkAcademy of Sciences
51, 627-659.[36] Park J.-M and T. C Lubensky, 1996, Interactions between membrane Inclusions on Fluctuating Membranes.
Journal de Physique I
6, 1217.[37] Kim K. S., J. Neu, and G. Oster, 1998, Curvature-mediated interactions between membrane proteins.
Biophys.J.
75, 2274.[38] Chou T., K. S. Kim and G. Oster, 2001, Statistical thermodynamics of membrane bending-mediated protein-protein attractions.
Biophys. J.
80, 1075.[39] Lewandowski, E. P., J. A. Bernate, P. C. Searson, P C and K. J. Stebe, 2008, Rotation and Alignment ofAnisotropic Particles on Nonplanar Interfaces,
Langmuir , 24, 9302.[40] Lewandowski, E. P., J. A. Bernate, A. Tseng, P. C. Searson, P C and K. J. Stebe, 2009, Oriented assemblyof anisotropic particles by capillary interactions,
Soft Matter , 5, 886.[41] Frost A., R. Perera, A. Roux, K. Spasov, O. Destaing, E.H. Engelmann, P. De Camilli, and A, Unger, 2008,Structural Basis of Membrane Invaginations by F-BAR Domains.
Cell
J.Phys. France
50, 1557-1571.[43] Frank J. R. and M. Kardar, 2008, Defects in nematic membranes can buckle into pseudospheres,
Phys. Rev.
E 77, 041705.[44] Kumar P. B. S. and Madan Rao, 1998, Shape Instabilities in the Dynamics of a Two-Component FluidMembrane,
Phys. Rev. Lett.
80, 2489.[45] Kumar P. B. S., Gerhard Gompper and Reinhard Lipowsky, 2001, Budding Dynamics of MulticomponentMembranes,
Phys. Rev. Lett.
86, 3911.[46] Henriksen J.R., T.L. Andresen, L. Feldborg, L. Duelund, and J.H. Ipsen, 2010, Understanding DetergentEffects on Lipid Membranes: A Model Study.
Biophy. J.
98, 2199-2205.[47] Gerbeaud C, 1998, Effect de l’insertion de prot´eines et de peptides membranaires sur les propri´et´es m´ecaniqueset les changements morphologiques de v´esicules g´eantes.
Ph.d. thesis L’Universte Bordeaux I. [48] Gouttenoire J., P. Roingeard, F. Penin, and D. Moradpour1, 2010, Amphipathic α -Helix AH2 Is a MajorDeterminant for the Oligomerization of Hepatitis C Virus Nonstructural Protein 4B. Journal of Virology
Curr. Opinion in Cell Biol.
20, 427-436[50] Sweitzer S.M. and J.E. Hinshaw, 1998, Dynamin undergoes a GTP-dependent conformational change causingvesiculation.
Cell
93, 1021.[51] Sorre B, A. Callan-Jones, J. Manzi, B. Goud, J. Prost, P. Bassereau, A. Roux, 2012, Nature of curvaturecoupling of amphiphysin with membranes depends on its bound density.
Proc. Natl. Acad. Sciences
J. Phys. France II
2, 371-382.[53] Vitelli V. and D.R. Nelson, 2004, Defect generation and deconfinement on corrugated topographies.
Phys.Rev.
E 70, 051105.[54] Ramakrishnan N., J. H. Ipsen, P. B. Sunil Kumar, 2012, Role of disclinations in determining the morphologyof deformable fluid interfaces.
Soft Matter
8, 3058-3061.[55] Roux, A., G. Koster, M. Lenz, B. Sorre, J.-B. Manneville, P. Nassoy and P. Bassereau, 2010, Membranecurvature controls dynamin polymerization.
Proc. Natl. Acad. Sciences
Journal of Structural Biol.
Nature Cell Biology
3, 922-926.[58] Roux A., K. Uyhazi, A. Frost, and P. De Camilli, 2006, GTP-dependent twisting of dynamin implicatesconstriction and tension in membrane fission.
Nature
44, 528-531.[59] Itoh T. and P. De Camilli, 2006, BAR, F-BAR(EFC), and ENTH/ANTH domains in the regulation ofmembrane cytosol interfaces and membrane curvature.
Biochim. Biophys. Acta
60] Shibata Y., C. Voss, J.M. Rist, J. Hu, T.A. Rapoport, W.A. Prinz and G.K. Voeltz, 2008, The reticulonand DP1/Yop1p proteins form immobile oligomers in the tubular endoplasmic reticulum.
J. Biol. Chem.283,18892-904.