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Cite this: SoftMatter, 2017,

13, 4099

Membrane structure formation induced by twotypes of banana-shaped proteins†

Hiroshi Noguchi *a and Jean-Baptiste Fournier*b

The assembly of banana-shaped rodlike proteins on membranes and the associated membrane shape

transformations are investigated by analytical theory and coarse-grained simulations. The membrane-

mediated interactions between two banana-shaped inclusions are derived theoretically using a point-like

formalism based on fixed anisotropic curvatures, both for zero surface tension and for finite surface

tension. On a larger scale, the interactions between the assemblies of such rodlike inclusions are

determined analytically. Meshless membrane simulations are performed in the presence of a large

number of inclusions of two types, corresponding to the curved rods of opposite curvatures, both for

flat membranes and vesicles. Rods of the same type aggregate into linear assemblies perpendicular to

the rod axis, leading to membrane tubulation. However, rods of the other type, those of opposite

curvature, are attracted to the lateral sides of these assemblies, and stabilize a straight bump structure

that prevents tubulation. When the two types of rods have almost opposite curvatures, the bumps

attract one another, forming a striped structure. Positive surface tension is found to stabilize stripe

formation. The simulation results agree well with the theoretical predictions provided the point-like

curvatures of the model are scaled-down to account for the effective flexibility of the simulated rods.

1 Introduction

In living cells, membrane shape transformation plays a key rolein biological functions such as endo/exocytosis and vesicletransports. Cell organelles have specific shapes depending ontheir functions. Various types of proteins participate in theregulation of these dynamic and static membrane shapes.1–6

These proteins mainly control local membrane shapes in twoways: hydrophobic insertions (wedging) and scaffolding. In theformer mechanism, a part of the protein, such as an amphi-pathic a-helix, is inserted into the lipid bilayer membrane. Inthe latter mechanism, the protein domain has a strong affinityfor the lipid polar head groups and adsorbs onto the lipidmembrane. A BAR (Bin/Amphiphysin/Rvs) domain, which con-sists of a banana-shaped dimer, mainly bends the membranealong the domain axis via scaffolding.7–11 Some of the BARsuperfamily proteins, such as N-BAR proteins, also have hydro-phobic insertions. Experimentally, the membrane tubulationand curvature-sensing by various types of BAR superfamilyproteins have been observed.7–22

Objects with rotational symmetry, such as spherical colloidsor conical integral proteins inserted perpendicularly to themembrane, generate an isotropic membrane curvature. Conversely,BAR domains which are banana-shaped generate an anisotropiccurvature23–26 (amphipathic a-helices can also yield an anisotropiccurvature27). Theoretical models have shown that membraneinclusions, such as adsorbed or embedded proteins, or colloids,undergo long-range interactions that are mediated by the curva-ture elasticity of the membrane.28 It was also shown that aniso-tropic inclusions experience interactions of a longer range thanisotropic ones and are able to produce complex aggregates.29,30

To simplify the theoretical calculations, membrane inclusionsare usually modelled as non-deformable objects with a fixedcurved shape.30–33 Owing to their small sizes it is often con-venient to treat them as point-like objects.30,31,34 Although theexistence of membrane-mediated interactions has been verifiedexperimentally in the case of isotropic inclusions,35 there is nodirect experimental evidence yet in the case of anisotropicproteins, despite their biological importance.

Numerical simulations are therefore essential in this context.Atomic and coarse-grained molecular simulations36–40 have beenemployed to investigate molecular-scale interactions betweenBAR proteins and lipids. The scaffold formation38 and linearassembly39 of BAR domains have been demonstrated. To investigatelarge-scale membrane deformations, a dynamically triangulatedmembrane model41,42 and meshless membrane models43–47

have been employed; consequently, various (meta)stable vesicle

a Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581,

Japan. E-mail: noguchi@issp.u-tokyo.ac.jpb Laboratoire Matiere et Systemes Complexes (MSC), UMR 7057 CNRS,

Universite Paris Diderot, F-75205, Paris, France.

E-mail: jean-baptiste.fournier@univ-paris-diderot.fr

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7sm00305f

Received 13th February 2017,Accepted 11th May 2017

DOI: 10.1039/c7sm00305f

rsc.li/soft-matter-journal

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shapes41–44 and the tubule formation dynamics45 have beenreported. However, the relation with the large-scale pictureof the theory of point-like anisotropic inclusions30–32 is notwell investigated. Here, we compare the meshless membranesimulations with this theory.

In living cells, more than one type of BAR and other proteinscooperatively work to regulate membrane shape. However,in most of the previous reports, rods of a single type areconsidered in theories and simulations. To our knowledge,only two studies have been reported for the interactions of twotypes of proteins. The mixture of inclusions with isotropic andanisotropic curvatures was shown to produce the self-assemblyof a neckless of anisotropic inclusions around a domainconsisting of a lattice of isotropic inclusions, mimicking theassembly of dynamin proteins around a scaffolded bud.32 Thephase segregation of rods with different positive spontaneouscurvatures is reported in ref. 42.

In the present study, we investigate the membrane-mediatedinteractions between two types of protein rods with oppositespontaneous curvatures, theoretically and numerically. Thiscorresponds to the situations in which oppositely curved proteins,such as I-BAR and other BAR proteins, are adsorbed on the sameleaflet of a bilayer membrane, or alternatively, two types ofpositively curved proteins are adsorbed on opposite leaflets.

In our theoretical analysis we use a multi-scale approach.First we consider the interaction between two rods of possiblydifferent curvatures. We assume that the rods are separated by adistance larger than their size so that we can treat them as point-like anisotropic inclusions using the Green function formalism ofref. 31 and 32. Contrary to previous models30–32 we assume here,in order to model banana-shaped BAR domains, that the inclu-sions fix a curvature in the rod’s direction but do not imposeany curvature in the orthogonal direction. We also consider theinteraction under nonzero surface tension, contrary to what wasdone in previous work.30–32 Our results show that rods of the samecurvature self-assemble into long straight structures that attractrods of opposite curvature on their sides. We then change thescale of our analysis and study the interaction between thesemacroscopic straight structures. We find that their interactiondepends crucially on membrane tension.

In our simulations an implicit-solvent meshless membranemodel43–46,48–50 is used to represent a fluid membrane. Banana-shaped proteins, assumed to be strongly adsorbed onto themembrane, are modelled together with the membrane regionbelow them as linear strings of particles with a bendingstiffness and a preferred curvature. In order to investigate themembrane-mediated interactions, no direct attractive interactionis considered between the rods. We investigate the interaction andstructures produced by a mixture of a large number of rods of twotypes. We find that the results of our simulations agree very wellwith the theoretical predictions provided the point-like curvaturesare scaled-down to account for the rod flexibility.

In Section 2, our multi-scale theoretical analysis of theinteractions between curved rods adsorbed onto a membranewith bending rigidity and tension is presented. In Section 3, thesimulation model and method are described. In Section 4, the

simulation and theoretical results are compared for the inter-actions of two rods. In Sections 5 and 6, the assembly of proteinrods in flat membranes and vesicles is presented. The summaryand discussion are given in Section 7.

2 Theory2.1 Interactions between two curved rods

We consider BAR-like membrane inclusions shaped as rods oflength rrod that are curved in a plane perpendicular to the planeof the membrane. In order to compute the interaction betweensuch inclusions, at separations well larger than rrod, we modelthem as point-like inclusions that impose some membranecurvature C along the rod direction and no constraint alongthe orthogonal direction. To derive the interaction between twosuch rods we use the Green function formalism of ref. 31 and 32.We first discuss the case of a tensionless membrane, then wetake into account membrane tension. In this model, in order tomake the calculations tractable, we neglect two aspects: thefinite length of the rods and their effective flexibility. However,as we shall see, these assumptions are not critical.

2.1.1 Tensionless membrane. We first consider two suchrods adsorbed onto a membrane with vanishing tension (g = 0).Let r12 be the vector going from the center of rod 1 to the centerof rod 2, and r21 = �r12. We assume that the rods imposecurvatures Cr1 and Cr2 along the directions u1 and u2, atangles y1 and y2 with respect r12 and r21, respectively (Fig. 1).To preserve the symmetry between the two inclusions, we orienty1 counterclockwise and y2 clockwise. We call R = |r12| thedistance between the rods.

We consider the limit of small membrane deformations.Minimizing the Helfrich bending energy of the membrane51

with the curvature constraints yields an interaction which is abinary quadratic form in Cr1 and Cr2, as shown in Appendix A.We find that unless |Cr1| { |Cr2|, or the opposite, the inter-action is well approximated by the sole term proportionalto Cr1Cr2 up to distances comparable to rrod. Accordingly, inthe following we neglect the contributions proportional to Cr1

2

and Cr22. The curvature-mediated interaction energy between

two rods in the absence of membrane tension, Hint(R), is thusobtained as

~HintðRÞ ’ ~Hð0Þint ðRÞ

¼ 16prrod4

9R2kCr1Cr2 cos 2y1ð Þ þ cos 2y2ð Þ � cos 2y1 � 2y2ð Þ½ �:

(1)

Fig. 1 Geometrical parameters for two rodlike inclusions.

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Here, k is the bending rigidity of the membrane. We haveexhibited the leading order term p1/R2, which describes theinteraction quantitatively well for R larger than a few times rrod.Note that this interaction is of longer range than the p1/R4

fluctuation-induced Casimir interaction between straight rodsand it has a different angular dependence.52,53 Graphs of thisinteraction are represented in Fig. 2a–c for various orientationsand curvatures of the rods.

When one of the rods is parallel to the separation vector tothe other rod (y1 = 0 or y2 = 0), eqn (1) gives

~Hð0Þint ðRÞ ¼

16prrod4

9R2kCr1Cr2; (2)

which is independent of the orientation of the other rod.

In contrast, when the axes of both rods are perpendicular tothe separation vector, i.e., y1 = y2 = p/2, H(0)

int has the oppositesign and an amplitude three times larger:

~Hð0Þint ðRÞ ¼ �

16prrod4

3R2kCr1Cr2: (3)

Hence, when two rods are identical, i.e., Cr1 = Cr2, they have astrong attractive interaction at y1 = y2 = p/2, while they have aweak repulsive interaction at y1 = 0 or y2 = 0. Obviously, whenthey are perpendicular to their separation vector they willattract up to contact since they match exactly and produce lessdeformation when they superimpose.

When two rods have curvatures of opposite sign, i.e., Cr1Cr2 o 0,the interactions are opposite. For y1 = 0 or y2 = 0, the rods havea weak attractive interaction. Rods of opposite curvatures

Fig. 2 Normalized interaction energy between two rods at vanishing tension, Hint (gray curves, plain and dashed), and at non-vanishing tension, Hint

(black curves, plain and dashed). Left column: Comparison between the interaction Hint at zero tension (gray) and the interaction Hint at a strong tensioncorresponding to x/rrod = 1 (black). Right column: Logarithmic plot of the interaction at weak tension, for x/rrod = 20, showing the crossover at R E 3xfrom the zero-tension behavior BH(0)

int to the nonzero-tension behavior BH(1)int. Top: Side-to-side orientation of the rods, i.e., y1 = y2 = p/2. Middle: Tip-to-

tip orientation of the rods, i.e., y1 = y2 = 0. Bottom: Tip-to-side orientation of the rods, i.e., y1 = 0 and y2 = p/2.

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in a tip-to-tip configuration produce no large-scale curvaturedeformation when they get close, therefore they will attractup to contact. This will likely be true also for rods not exactlyopposite in curvature.

Thus, for neighboring rods at equilibrium, we expect identicalrods to align preferentially side-to-side and rods of oppositecurvature to align tip-to-tip.

2.1.2 Membrane under tension. The derivation of the inter-action in the presence of membrane tension, Hint(R), is describedin Appendix A, Section A.2. For the same reasons as previously, weretain in the interaction only the terms proportional to Cr1Cr2. Inthe regime of strong tensions, i.e., for x = (k/g)1/2 E rrod, Fig. 2a–c,show that the interaction Hint displays several similitudes anddifferences with the tensionless case:

1. At short distances the attractive/repulsive behavior isthe same as in the tensionless case. Hence, identical rods willbind side-to-side and rods of opposite curvature tip-to-tip, asshown previously.

2. The range of the interaction is shorter in the presence oftension: the black curves decay more rapidly that the gray ones.

3. The interaction in the tip-to-tip configuration is notmonotonic in the strong tension case (see Fig. 2b, black lines).

It is well known that tension effects are negligible on length-scales smaller than x while they dominate on length-scaleslarger than x. Accordingly, in the weak tension regime, the fullinteraction exhibits a crossover at R E 3x from a bending-dominated regime to a tension-dominated regime:

HintðRÞ ¼

~Hð0Þint ðRÞ; for rrod oR� x;

Hð1Þint ðRÞ; for R� x if cos 2 y1 � y2ð Þ½ �a0;

Hð2Þint ðRÞ; for R� x if cos 2 y1 � y2ð Þ½ � ¼ 0;

8>>>><>>>>:

(4)

where

Hð1Þint ðRÞ ¼ �

64prrod4x2

3R4kCr1Cr2 cos 2 y1 � y2ð Þ½ �; (5)

Hð2Þint ðRÞ ¼

2ffiffiffi2p

p3=2rrod4

9x3=2e�R=xffiffiffiffi

Rp kCr1Cr2

� 2þ 2 cos 2y1ð Þ þ 2 cosð2y2Þ þ cos 2y1 þ 2y2ð Þ½ �:(6)

In Fig. 2d–f, one can see the corresponding crossover from aB1/R2 power-law to a B1/R4 power-law. Note that the asymp-totic interaction H(1)

int(R) depends only on the relative orientationof the inclusions, not in the direction of the separation vector.For particular orientations such that cos[2(y1 � y2)] = 0 (graphnot shown), the B1/R4 power-law disappears and is replaced bythe exponential decay H(2)

int(R).

2.2 Interactions between rod assemblies

Since identical rods attract strongly and align side-to-side, whileopposite rods align preferentially tip-to-tip, we expect the followingscenario. Identical rods should aggregate into straight rodassemblies, and parallel rod assemblies should either repel

one another, if they are made of rods of the same curvature, orattract one another side-to-side if they are made with rods ofopposite curvatures. Alternate and periodic striped structuresare therefore expected to develop.

We detail in Appendix B the calculation of the interactionFint(R) between two coarse-grained parallel rod assemblies oflength L that are separated by a center-to-center distance R. Theresult is

FintðRÞ ¼L

2

ffiffiffiffiffikgp

rrod2Cr1Cr2e

� R�rrodð Þ=x þ Ftip; (7)

where Ftip is the contribution coming from the extremities ofthe rods, which is subdominant and which we do not evaluate.

This interaction confirms that parallel rod assemblies of likecurvatures repel while rod assemblies of opposite curvaturesattract. The first term, proportional to the length of the rods,vanishes in the absence of tension, i.e., for g = 0. This can easilybe understood as flat membrane patches can fit in between therod assemblies at no cost in the absence of tension. However,some interaction will originate from the extremities of therod assemblies (Ftip), similar to that of isolated rods. In thepresence of tension, the energy per unit length will dominate andproduce the attraction of rod assemblies of opposite curvature.We see then two properties: (1) the interaction is short-ranged,decaying over x, and (2) the larger the tension the stronger theinteraction.

3 Simulation model and method

Since the details of the meshless membrane model and proteinrods are described in ref. 50 and ref. 43, 46, respectively,we briefly describe the model here. A fluid membrane isrepresented by a self-assembled one-layer sheet of N particles.The position and orientational vectors of the i-th particle are ri

and ti, respectively. The membrane particles interact with eachother via a potential U = Urep + Uatt + Ubend + Utilt. The potentialUrep is an excluded volume interaction with a diameter s for allpairs of particles. The solvent is implicitly accounted for by aneffective attractive potential Uatt.

The bending and tilt potentials are given by Ubend=kBT ¼kbend=2ð Þ

Pio j

ti � tj � Cbdri;j� �2

wcvðri;jÞ and Utilt=kBT ¼

ðktilt=2ÞPio j

ti � ri;j� �2þðtj � ri;jÞ2h i

wcv ri;j� �

, respectively, where

ri,j = ri � rj, ri,j = |ri,j|, ri,j = ri,j/ri,j, wcv(ri,j) is a weight function,and kBT denotes the thermal energy. The spontaneous curva-ture C0 of the membrane is given by C0s = Cbd/2.50 In this study,C0 = 0 and kbend = ktilt = 10 except for the membrane particlesbelonging to the protein rods.

A protein rod is modeled as a linear chain of Nsg membraneparticles. We use Nsg = 10, which corresponds to the typicalaspect ratio of the BAR domains. The BAR domain width isapproximately 2 nm, and the length ranges from 13 to 27 nm.8

Two types of protein rods, called as rods 1 and 2, are used.As detailed below, rods 1 and 2 have positive and negativespontaneous curvatures Cr1 and Cr2 along the rod axis, respectively,

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and both rods have no spontaneous curvature perpendicular tothe rod axis. Hereafter, we call the membrane particles forminga protein rod protein particles. The protein particles in eachprotein rod are further connected by a bond potential Urbond/kBT = (krbond/2s2)(ri+1,i � lsg)2 where krbond = 40, krbend = 4000,and lsg = 1.15s are used. The bending potential is given byUrbend/kBT = (krbend/2)(ri+1,i�ri,i�1� Cb)2, where Cb = 1 � (Cr1lsg)2/2and Cb = 1� (Cr2lsg)2/2 for rods 1 and 2, respectively. For bondedpairs of protein particles, we use four times larger values of kbend

and ktilt, together with the corresponding spontaneous curvature,to prevent the rod from bending tangentially in the membrane.

We employ the parameter sets used in ref. 43. The membranehas mechanical properties that are typical of lipid membranes:bending rigidity k/kBT = 15 � 1, the area of the tensionlessmembrane per particle a0/s2 = 1.2778 � 0.0002, the area of thecompression modulus KAs

2/kBT = 83.1 � 0.4, and the edge linetension Gs/kBT = 5.73 � 0.04. This edge tension G is sufficientlylarge to prevent membrane rupture in this study.46 Moleculardynamics using a Langevin thermostat is employed.50,54 In thefollowing, the results are displayed with the rod length rrod = 10sfor the length unit, kBT for the energy unit, and t = rrod

2/D for thetime unit, where D is the diffusion coefficient of the membraneparticles in the tensionless membrane.46

For flat membrane simulations, the NgLzT ensemble underperiodic boundary conditions is used. The projected area Axy =LxLy is fluctuated for a constant surface tension g while main-taining the aspect ratio Lx = Ly.55,56 To investigate the paircorrelation of the rods, two rods are set on a flat membranewith N = 6400 and the distance rgg of the centers of the mass ofthe rods are constrained by a harmonic potential kgg(rgg �rrod)2/2, where kgg = 10kBT/s2. The normalized rod end-to-endvector is used to determine the rod orientation.

For self-assembly, the rod-1 curvature is fixed as Cr1rrod = 4,which can induce membrane tubulation with a circumferenceof C2rrod for the present bending elastic constants. Flat mem-branes with N = 25 600 are investigated with various valuesof Cr2 = �crCr1. The density of the rods 1 and 2 are set tofr1 = Nr1Nsg/N = 0.1 and fr1 = Nr2Nsg/N = 0.2, where Nr1 and Nr2

are the numbers of the rods 1 and 2, respectively.For the rod assembly on a vesicle, the NVT ensemble is used

at N = 9600 and fr1 = fr2 = 0.25. The radius of the vesicle isRves = 3.07rrod in the absence of the rods. For the annealingsimulations, the rod curvatures are changed from Cr1 = Cr2 = 0to Cr1rrod = 4 while maintaining the ratio Cr2 = �crCr1 with theannealing rate Cr1/dt = 0.02/rrodt and subsequently the vesicle isequilibrated for 1500–2500t. For each simulation condition, thereproducibility is confirmed at least from four different initialconformations.

4 Comparison of theory andsimulation on the interaction oftwo rods

Before investigating the self-assembly of the rods, we comparethe simulation results with the theoretical predictions in the

tensionless case. Due to the smallness of the interactions, it isdifficult to determine numerically the potential of the meanforce between two rods. Instead, we compute the angulardistributions and angular correlations for an isolated pair ofrods separated by the short distance R = rrod. We thus deter-mine the statistics of the following two quantities S1 = cos2(y1)and S2 = sin2(y1)sin2(y2), as they turn out to best capture theangular behaviors (see Fig. 1 for the definition of the angles).

The theory predicts P(y1, y2) = exp(�Hint/kBT)/Z, with Hint

given by eqn (1), where Z is the normalization factor forÐ 10P Sað ÞdSa ¼ 1 (a = 1 or 2). For rods with zero curvatures,

Hint = 0, thus P(y1) = 1/(2p) and P S1ð Þ ¼ 1�pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS1 1� S1ð Þ

p� �. The

simulations confirm this dependence, as shown by the greenand superimposed dashed lines in Fig. 3a and b. This showsthat if there is a Casimir-like fluctuation-induced interaction(neglected in our calculations), arising from the contrast ofrigidity between the rods and the surrounding membrane,it must be very weak.

We consider now curved rods. When two rods have equalcurvatures, the theory implies that the orientations u1 and u2

are preferentially perpendicular to r12 and therefore smallervalues of S1 become more probable. Conversely, for rods ofopposite curvatures, u1 and u2 are preferentially parallel to r12

and thus larger values of S1 become more probable. Thesetrends are confirmed by the simulations, as evidenced by the

Fig. 3 Probability distribution of the angular parameters S1 (a) and S2 (b)for two rods separated by R = rrod at zero surface tension (g = 0). Thecolored solid lines represent the simulation data for identical positiverod curvature Cr1 = Cr2 = 4/rrod (red, labeled by pp), opposite curvaturesCr1 = �Cr2 = 4/rrod (cyan, labeled by pn), and zero curvature Cr1 = Cr2 = 0(green, labeled by 0). The dashed lines are deduced from Hint in eqn (1), takinga prefactor of 3kBT, �3kBT, and 0 for pp, pn, and 0, respectively. A typicalsnapshot is shown in the inset of (a). The red and blue particles represent therod segments and the gray particles represent membrane particles.

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red and cyan lines in Fig. 3a. For the same reasons, large valuesof S2 have larger probability for rods of equal curvatures andsmaller probability for rods of opposite curvatures, in agree-ment with Fig. 3b. With the parameters k, rrod and R givenpreviously, we obtain a good fit of the numerical results bythe theory provided we renormalize the curvatures by a factorC1/20 (see the agreement between the dashed lines and thecolored solid lines in Fig. 3a and b). This apparently large factoris reasonable given the amplitude of the rod shape fluctuationsin Fig. 3a, and it is probably due to the finite values ofkrbond and krbend and to the large number of particles in a rod(Nsg = 10). We conclude that our model captures quantitativelythe angular dependence and the amplitude of the rod inter-action provided renormalized curvatures are used.

5 Rod assembly on the flat membrane

First, we describe the rod assemblies on the tensionlessflat membrane (g = 0). Fig. 4 shows typical snapshots. For zerorod-curvature of rods 2 (cr = 0), rods 1 form disk-shaped tubuleslike in the absence of rods 2. Rods 2 are distributed in the flatregion of the membrane. Small tubules have a semicircular-disk shape. In contract, large tubules have a disk shape (like amussel shell) connected to the flat membrane by a narrowcylindrical neck (see the upper tubule in Fig. 4a). We believethat these disk-like structures could be the microscopic pre-cursors of the tubules induced by BAR proteins in experiments.Like in other endocytotic processes (e.g., clathrin-mediatedendocytosis) the rods shaping these tubular vesicles are at thesame time their cargos. This assembly of the rods 1 agrees with thetheoretical prediction for the point-like inclusions in Section 2.1(identical curved rods attract and align side-to-side). Identicalrods have an attractive interaction perpendicular to the rod axes.The neighboring rods 1 contact each other and subsequently

many rods 1 assemble and form a straight one-dimensional bandcurved along the short axis of the assembly (parallel to the rodaxis). Eventually, long band assemblies bend along their long axisand form a tubule, as this reduces the bending energy costaround the assembly. Rods 2, with Cr2 = 0, exhibit little interac-tions with rods 1. This also agrees with the prediction of eqn (1).

As the cr increases, the necks of the large tubules remainopen. For cr = 0.25, the rod-1 assemblies exhibit wide walls orhill shapes (see Fig. 4b). Rods 2 are concentrated around thefoot of these walls and stabilize their negative curvature. Uponfurther increasing the cr, rods 2 more clearly assemble on bothsides of the rod-1 assemblies and form long straight bumps [seeFig. 4c and the corresponding movie provided in the ESI†(Movie 1)]. These structures recall the shape of a centipede ora myriapod: the rod-1 straight assembly looks like the body of acentipede and rods 2 on both sides are like many legs. Thisattraction of oppositely curved rods along their axis also agreeswith the theoretical prediction for the point-like inclusions inSection 2.1 (oppositely curved rods attract and align tip-to-tip).

The straight assemblies are also stable at larger cr. However,at cr \ 0.58, straight assemblies of opposite curvatures haveattractive interaction in the lateral direction (parallel to the axisof each rod) and form a periodic bump structure (see Fig. 4d).Each rod-1 (-2) straight assembly is connected to two rod-2 (-1)straight assemblies on both lateral sides. We call this assemblyof rods 1 and 2 a stripe assembly, since it forms periodic bands.This stripe assembly is metastable but has a longer life timethan our simulation periods at cr = 0.75. Even when the stripeassembly is set as an initial conformation, the rod-1 straightassemblies separate to form isolated bumps at cr t 0.5.

Fig. 5 shows the assembly dynamics at cr = 0.625 [the corres-ponding movie is provided in the ESI† (Movie 2)]. First, shortstraight assemblies are formed. As the tips of the bumps approachclosely, they fuse into one large straight assembly. In contrast, asthey approach laterally, they form a stripe assembly of shortstraight assemblies. As a long bump approaches, the stripeassemblies are disassembled and reassemble into longer assem-blies (see the upper region of the three right snapshots in Fig. 5).Note that some rods of type 2 also contact the rod-1 assemblies atthe tips of the assemblies (see snapshots at t/t = 12.5 and 25 inFig. 5), so that an attractive interaction exists between the rods oftypes 1 and 2 even when the rod-1 axis is perpendicular to theseparation vector. This attraction is in good agreement with theprediction of eqn (1) with y1 = 0 and y2 = p/2.

As the rods assemble, the projected membrane area Axy

decreases while the mean vertical rod span zrm and the meancluster size Ncl of rods 1 increases (see Fig. 6 and 7). At cr Z 0.625,the rod-1 cluster formation becomes slower than at cr = 0.5,since the stripe assembly suppresses the fusion between rod-1assemblies. The vertical span is calculated from the height variance of

all rod segments as zrm2 ¼

Pi2rods

zi � zGð Þ2.

Nr1 þNr2ð ÞNsg where

zG ¼P

i2rodszi�Nr1 þNr2ð ÞNsg. A rod is considered to belong to a

cluster when the distance between the centers of mass of the rodand one of the rods in the cluster is less than rrod/2. For rods 1,

Fig. 4 Snapshots of assembled protein rods on the membrane at thesurface tension g = 0. (a) The ratio of the rod curvatures, cr = �Cr2/Cr1 = 0.(b) cr = 0.25. (c) cr = 0.5. (d) cr = 0.75. The red and blue particles representthe segments of rods 1 and 2, respectively. The gray particles representmembrane particles.

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the mean cluster size is given by Ncl ¼PNr1

icl¼1icl

2ncli

!,Nr1 with

Nr1 ¼PNr1

icl¼1icln

cli where ncl

i is the number of clusters of size icl.

At small positive surface tensions (x c rrod, i.e., g { gc =k/rrod

2), the bending energy contribution should be dominant

in the interactions between the rods as described in Section 2. Oursimulation results support this theoretical prediction. The rodassembly is only slightly modified at grrod

2/kBT t 5 (x \ 1.7rrod).In contrast, a large positive surface tension (g \ gc) suppressesboth the decrease of the projected area Axy and the protrusion ofthe rod assemblies (see Fig. 6). Since the curvature-energy gain byrod assembling is reduced when increasing g (see the theory inSection 2.1.2 and Fig. 2), the rod cluster size decreases and morerods remain in an isolated state (see Fig. 7b and 8). Interestingly,the stripe assembly is stabilized for smaller cr at larger values ofg, as shown in Fig. 9. For cr = 0.5, the stripe assembly exists forgrrod

2/kBT \ 50 while it does not for grrod2/kBT t 50 (compare

Fig. 4c and 8). This agrees well with the theoretical results ofSection 2.2 showing that the attraction between long straightassemblies of opposite curvature scales as Bg1/2 and thusincreases with membrane tension.

6 Vesicle deformation by rod assembly

The rods also assemble on vesicles like on flat membranes.However, the assembly structures are largely modified by the

Fig. 5 Sequential snapshots of the self-assembly of protein rods at therod-curvature ratio cr = 0.625 and the surface tension g = 0.

Fig. 6 Time evolution of (a) the projected membrane area Axy and (b)mean vertical rod span zrm for the rod-curvature ratio cr = 0.5, 0.625, and0.75 at the surface tension grrod

2/kBT = 0 and 50. At cr = 0.625 and g = 0,the same data are used as in Fig. 5.

Fig. 7 Time evolution of mean cluster size of the rods 1. (a) The rod-curvature ratio cr = 0.5, 0.625, and 0.75 at the surface tension g = 0.(b) grrod

2/kBT = 0, 5, 50, 100, and 200 at cr = 0.5. The same data are used asin Fig. 6 for the corresponding parameter sets.

Fig. 8 Snapshots of the assembled protein rods on the membrane at therod-curvature ratio cr = 0.5 for (a) the surface tension grrod

2/kBT = 100 and(b) grrod

2/kBT = 200.

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original vesicle curvature and its closed geometry. When a singletype of rod exists on the vesicle, the rods induce oblate or polyhedralshapes of the vesicle and they assemble at the edges of thepolyhedron.43,44 For negative rod curvatures, invaginations into theinside of the vesicle can occur.44,45 Here, we set the rod-1 density,fr1 = 0.25, to make the vesicle form a disk shape in the absence ofrods 2. We add rods 2 with the same density and investigate how thevesicle shape is changed upon varying cr. Note that the vesiclesmaintain the spherical topology without membrane rupture.

Fig. 10a and b show the snapshots obtained by annealingfrom Cr1 = Cr2 = 0 to Cr1rrod = 4 at a fixed ratio cr. For cr = 0.375,the vesicles form a disk shape with a concave circular regionbetween the center and the rim; rods 2 are aligned along theradial directions of the disk in this concave region. For cr = 1,rods 2 form invaginations into the inside of the vesicle and the

two rims of the rod-1 assembly are partially connected by therod-2 assembly like in the flat membrane. For intermediatevalues of cr, the obtained vesicle shapes depend on the initialconformations and thermal noise during annealing: somevesicles have an invagination or partial doubled rims.

To investigate the stability of the disk-shaped and invaginatedvesicles, we simulated vesicles with different cr starting from theannealed conformation at cr = 0.375 and 1 as initial states. As the cr

increases for disk-shaped vesicles, the center region of the vesiclebecomes more spherical. This shape change reduces the radiusof the disk, resulting in a buckling or winding of the disk rim, atcr = 0.625 and 0.75 (see Fig. 10c). At cr = 0.875 and 1, rods 2 form astraight dimple on the side of the rod-1 assembly (as seen in the leftregion of the sliced snapshot in Fig. 10b) as well as an invagination.

Starting from an invaginated vesicle, the invagination swellsto an ellipsoidal shape and the rest of the rods 1 form a circularrim, at cr = 0.5 (see Fig. 10d). At cr = 0.375, the invagination isremoved and a disk-shaped vesicle is obtained [see the movieprovided in the ESI† (Movie 3)].

These shape changes can be characterized by the radius ofgyration Rg and the shape parameter, asphericity asp (seeFig. 10e and f). The asphericity is the degree of deviation froma spherical shape and is expressed as57

asp ¼l1 � l2ð Þ2þ l2 � l3ð Þ2þ l3 � l1ð Þ2

2Rg4

; (8)

where Rg2 = l1 + l2 + l3 and l1 r l2 r l3 are the

eigenvalues of the gyration tensor of the vesicle:

Fig. 9 Phase diagram of the rod assembly for the surface tension g andthe ratio of the rod curvatures cr.

Fig. 10 Vesicle deformation by oppositely curved rods at various rod-curvature ratios cr. (a–d) Snapshots. The membrane particles are represented by atransparent gray sphere. The rod 1 (resp. rod 2) segments are represented by a sphere half red and half yellow (resp. half blue and half green). The orientationvector ti lies along the direction from the yellow (green) to red (blue) hemispheres. (a and b) Vesicles formed by annealing to Cr1rrod = 4 with (a) cr = 0.375 and(b) cr = 1. (c) Vesicles formed by an increase in cr from 0.375 to 0.75 at Cr1rrod = 4. (d) Vesicles formed by a decrease in cr from 1 to 0.5 at Cr1rrod = 4. Top andbottom panels show the snapshot of all particles in bird’s-eye view and a sliced snapshot in front view, respectively. (e and f) Dependence of (e) the mean radius ofgyration hRgi and (f) mean asphericity hasp i of vesicles on cr. The data labeled up and down are obtained by an increase and decrease in cr at Cr1rrod = 4.

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aab ¼Pj

aj � aG� �

bj � bG� ��

N, where a, b = x, y, z and aG are

the coordinates of the center of mass. For a perfect sphere,asp = 0, for a thin rod asp = 1, and for a thin disk, asp = 0.25. Startingfrom a disk-shaped vesicle (labeled ‘up’ in Fig. 10e and f), thevesicles have more spherical and less compact shapes than invagi-nated vesicles so that they take larger Rg and smaller asp. Thus, thevesicle shapes can be chosen from many metastable conformationsby using the hysteresis of cr variations. Our simulations demonstratea control method of the vesicle shapes by using hysteresis.

7 Summary and discussion

We have studied the assembly structures of binary banana-shapedprotein rods. Compared to the situation where only one type ofrod is present, the coexistence of two types of rods, with oppositecurvatures, induces a greater variety of membrane shapes. Here,using meshless membrane simulations, we demonstrated straightbumps and striped structures in flat membranes, and buckledrims and ellipsoidal invaginations in vesicles. More complicatedshapes such as vesicles with periodic bumps, like cartridge pleats,can be formed for larger vesicles. These structure formations canbe examined in experiments with two types of BAR proteinsabsorbed on the same side of the membrane or with a singletype of protein absorbed on both sides of the membrane. Theperiodic length of the striped structure is the sum of the long-itudinal lengths of two BAR domains (20 to 50 nm). It shouldtherefore be observable by electron microscopy. Our simulationsuggests that imposing a positive surface tension helps in obtain-ing the striped structure.

For rods of like curvatures, our theory predicts a side-to-sideattractive interaction and a tip-to-tip repulsion. For rods havingcurvatures of opposite sign our theory predicts the reverse, i.e.,a tip-to-tip attraction and a side-to-side repulsion. Our coarse-grained simulations revealed that identical rods assemble inthe side-to-side configuration and thus build straight rodassemblies; these assemblies attract the rods with oppositecurvatures in the tip-to-tip configuration thereby inducing aside-to-side alignment of the formed assemblies. The for-mation of those alternate assemblies, or striped structures,therefore agrees well with our theoretical predictions.

With increasing membrane tension, we theoretically foundthat the attractive interaction between opposite rods in thetip-to-tip configuration (which corresponds to the orientationsinvolved in the striped structures) is increased at short distancesand decreased at large distances (see Fig. 2b). Likewise, ourlarger-scale analysis of Section 2.2 predicts a short-range attrac-tion between opposite straight rod assemblies that increaseswith membrane tension. These theoretical results explain wellthe observed increased stability of the striped structures whenthe tension is increased and the shorter length of the rodassemblies in the thermalized system.

These theoretical predictions are supported by our simulationresults although the theory considers rigid rods with a fixed shapewhile the rods in the simulation are flexible. Our model capturesquantitatively the angular dependence and the amplitude of the rod

interaction provided renormalized curvatures are used. The realproteins are not completely rigid but their stiffness is not measured.The rod flexibility can be considered to result in an effectivelysmaller rod curvature, not altering the sign of the interactions inthe present assemblies. The protein stiffness, together with the finitesize of the rod inclusions, should however be taken into account forquantitative predictions of the assembly conditions.

Here, we consider the case where the protein rods have zerospontaneous (side) curvature perpendicular to the rod axis. Theprotein–membrane interactions including amphipathic-helix inser-tions can yield non-zero side curvature. The finite rod and sidecurvatures can induce egg-carton30,31 and network structures.45 Ifthe membrane bending in the perpendicular direction is muchstronger than that along the rod axis, the effective anisotropic axisof the curvature is perpendicular to the rod axis. In such a case, theside-to-side attraction of the anisotropic inclusions correspondsthe tip-to-tip attraction for the rod axes. The linear assembly of theN-BAR domains in the coarse-grained molecular simulation ofref. 39 may be understood by the effect of the strong side curvature.For a quantitative comparison of our results with experiments,the elastic parameters of each type of protein are required. Theestimation of protein stiffness and side curvature by atomic orcoarse-grained molecular simulation is important.

It is interesting also to note that we find good agreementbetween theory and simulations although our calculations areset in the linearized regime for membrane deformation, contraryto what was found for isotropic particles in the strong deforma-tion regime.24 Either anisotropic particles are more adapted tolinearized calculations, or the curvatures in the regimes wherethe assemblies take place are not so strong: further investiga-tions along these lines would be interesting.

The effects of the Casimir force, induced by the fluctuationsof the membrane, remain to be investigated further. It wasignored in the present work since it is normally sub-dominantfor small inclusions imposing large membrane deformations.This Casimir force is known however to induce an effectiveattraction between long straight rods.52,53 We have investigatedstraight rods in our simulations, and found that the latter havetoo weak Casimir interactions to induce rod clusters for thepresent length (Nsg = 10). However, twice longer rods (Nsg = 20)can form clusters. Thus, the assembly of longer rods can beinduced by both Casimir and rod-curvature forces.

In living cells, many types of BAR superfamily proteins coop-erate to regulate membrane shapes. Here, we have studied only themixture of two types of proteins. The cooperative effect of mixingmore than two proteins is an important topic for further studies.

Appendix A: interactions betweenpoint-like inclusions constraining themembrane curvature along a specificdirection

In the small deformation regime, the membrane shape can beexpressed in the Monge gauge by a height function z = h(r), with

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r = (x, y) a point of the projected plane. The Helfrich elasticenergy of the membrane is given, to quadratic order, by51

H ’ðd2r

k2r2h� �2þg

2rhð Þ2

h i¼ k

2

ðd2rhLh; (9)

with L = r4 � x�2r2, the operator associated with the

Hamiltonian, and x ¼ffiffiffiffiffiffiffiffik=g

pthe coherence length arising from

the membrane tension g and the membrane bending rigidity k.Let us consider a point-like inclusion, placed at r = 0, that

imposes a membrane curvature Cr1 along the direction given bythe unit vector u1 of angle y1 (see the definition of the anglesin Section 2.1), and a second point-like inclusion, placed atr = Rex, imposing a curvature Cr2 along the direction given bythe unit vector u2 of angle y2. We shall discuss later how tomathematically fix the ‘‘size’’ of these inclusions. The constraintsset by the inclusions, for small membrane deformations, read

h,11(0) h,xx(0)cos2 y1 + h,xy(0)sin(2y1) + h,yy(0)sin2 y1 = Cr1,(10)

h,22(Rex) h,xx(Rex)cos2 y20 + h,xy(Rex)sin(2y2

0)

+ h,yy(Rex)sin2y20 = Cr2, (11)

where a comma indicates differentiation. Here y20 is the polar

angle of the direction u2 with respect to the x-axis, i.e., y20 = p� y2.

Minimizing the Helfrich energy (9) with the above constraintsyields the Euler–Lagrange equation:

lh(r) = L1[d,xx(r)cos2 y1 + d,xy(r)sin(2y1) + d,yy(r)sin2 y1]

+ L2[d,xx(r � Rex)cos2 y20 + d,xy(r � Rex)sin(2y2

0)

+ d,yy(r � Rex)sin2 y20]. (12)

Here, d is the Dirac distribution, and L1 and L2 are Lagrangemultipliers. The solution is given by

h(r) = L1[G,xx(r)cos2 y1 + G,xy(r)sin(2y1) + G,yy(r)sin2 y1]

+ L2[G,xx(r � Rex)cos2 y20 + G,xy(r � Rex)sin(2y2

0)

+ G,yy(r � Rex)sin2 y20], (13)

where G(r) is the Green function defined by

lG(r) = d(r). (14)

A.1 Vanishing tension case

If g = 0, the Euler–Lagrange operator becomes l = r4 and theGreen function is given by28

GðrÞ ¼ r2

8pln r: (15)

Indeed, with rr2 = r�1qrrqr (cylindrical coordinates), the most

general rotationally symmetric solution of r2G(r) = 0 is G(r) =A1 + A2 ln r + A3r2 + A4r2 ln r. The finiteness of G at the originrequires A2 = 0 and integrating eqn (14) around r = 0 yieldsA4 = 1/(8p). The constants A1 and A3 are useful if one wishes tosatisfy some boundary conditions for large r, however they donot contribute to the fourth derivatives of G appearing below,so we can set A1 = A3 = 0, which yields eqn (15).

Satisfying the constraints (10) and (11) gives a linear set ofequations for the Lagrange multipliers:

ML = C, (16)

where L = (L1, L2)T, C = (Cr1,Cr2)T and M is a matrix such that

M11 = G,xxxx(0)cos4 y1 + 4G,xxxy(0)cos3 y1 sin y1

+ 6G,xxyy(0)cos2 y1 sin2 y1 + 4G,xyyy(0)cos y1 sin3 y1

+ G,yyyy(0)sin4 y1, (17)

M21 ¼ G;xxxx Rexð Þ cos2 y1 cos2 y20

þ 2G;xxxy Rexð Þ cos y1 cos y20sin y1 þ y2

0�

þ 1

4G;xxyy Rexð Þ 2þ cos 2y1 � 2y2

0�

� 3 cos 2y1 þ 2y20

� h i

þ 2G;xyyy Rexð Þ sin y1 sin y20sin y1 þ y2

0�

þ G;yyyy Rexð Þ sin2 y1 sin2 y20;

(18)

and with M12 obtained from M21 by replacing Rex by �Rex, andwith M22 obtained from M11 by replacing y1 by y2

0. Using eqn (9)and (12), one obtains exactly

H ¼ k2L1Cr1 þ L2Cr2ð Þ ¼ k

2CTM�1C: (19)

We need the fourth derivatives of the Green function. Fromeqn (15), we obtain

G;xxxx �Rexð Þ ¼ G;xxyy �Rexð Þ ¼ � 1

4pR2;

G;xxxy �Rexð Þ ¼ G;xyyy �Rexð Þ ¼ 0;

G;yyyy �Rexð Þ ¼ 3

4pR2;

(20)

It follows that the fourth derivatives of G(r), which appear inM11 and M22, are singular. This problem comes from the factthat we use mathematically a point-like constraint. Physically,the inclusions have a characteristic size, here a E rrod, andwe can specify this size by using an upper wavevector cutoffLE 1/a for the membrane deformation in the reciprocal space.We take numerically L = 2/a as in ref. 29 and 30, since thischoice was shown to match the exact calculations in the case ofisotropic inclusions. This gives

G;xxxxð0Þ ¼ð2=a0

qdq

ð2pÞ2ð2p0

dy cos4 y ¼ 3

8pa2: (21)

Similarly,

G;xxxyð0Þ ¼ G;xyyyð0Þ ¼ 0;

G;xxyyð0Þ ¼1

8pa2; and G;yyyyð0Þ ¼

3

8pa2:

(22)

We thus obtain

M11 ¼M22 ¼3

8pa2; (23)

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M12 ¼M21 ¼cos 2y1 � 2y2ð Þ � cos 2y1ð Þ � cos 2y2ð Þ

4pR2; (24)

where we have switched back to y2 = p � y20. Note that M12 is

symmetric upon the exchange of y1 and y2 as it should be.Using eqn (19), we obtain exactly

H ¼ 4pka24a2Cr1Cr2f y1; y2ð Þ þ 3R2 Cr1

2 þ Cr22

� �9R2 þ 4f y1; y2ð Þa4=R2

; (25)

where f (y1,y2) = cos(2y1) + cos(2y2) � cos(2y1 � 2y2). However,since the rods we are actually modeling are not point-like, theabove formula is not meaningful for R C a. We therefore takethe leading order of H in 1/R, which describes the interactionquantitatively well for R larger than a few times a. Replacinga by rrod yields the interaction H(0)

int(R) given by eqn (1). Notethat this result is symmetric upon exchanging y1 and y2 andinvariant in changing y1 into y1 + p or y2 into y2 + p, as itshould be.

A.2 Finite tension case

In the presence of membrane tension, for g a 0, the formalismin the previous section is still valid, provided the full Greenfunction is used, i.e., the Green function of l = r2(r2 � x�2)defined by eqn (14). In particular, eqn (17)–(19) still hold. TheGreen function in the presence of tension is given by58

GðrÞ ¼ � x2

2pK0

r

x

�þ ln r

� ; (26)

where K0 and I0 (used below) are modified Bessel functions.Indeed, the most general rotationally symmetric solution ofLG(r) = 0 is G(r) = A1 + A2 ln r + A3K0(r/x) + A4I0(r/x). Thefiniteness of G at the origin requires A2 = A3 and integratingeqn (14) around r = 0 yields A3 = �x2/(2p). Discarding theconstant term A1 and the term A4I0(r/x) that diverges at infinity,we obtain eqn (26). Note that eqn (26) reduces to eqn (15)for r { x.

Regularizing the fourth derivatives of the Green function aspreviously, we obtain

G;xxxxð0Þ¼ð2=a0

qdq

ð2pÞ2ð2p0

dyq4cos4yq4þx�2q2¼

3

8pa2� 3

32px2ln 1þ4x2=a2� �

:

(27)

Similarly,

G;xxxyð0Þ ¼ G;xyyyð0Þ ¼ 0;

G;xxyyð0Þ ¼1

8pa2� 1

32px2ln 1þ 4x2=a2� � (28)

G,yyyy(0) = G,xxxx(0). (29)

Using eqn (17)–(19) with these elements, we obtain the graphsof Hint(R) shown in Section 2.1.2 and the correspondingasymptotic behaviors H(1)

int(R) and H(2)int(R).

Appendix B: interactions betweenparallel straight rod-assemblies

We consider two parallel rod assemblies, one made of rods ofcurvature Cr1 and the other one made of rods of curvature Cr2.Recall that the rods are curved in the direction perpendicular tothe axes of the rod assemblies. Let us define r, g1 and g2 in sucha way that the curvatures Cr1 and Cr2 correspond to angularvariations 2g1 and 2g2 over the distance 2r = rrod, respectively(see Fig. 11). For the sake of simplicity, we regard the rodassemblies as infinite and homogeneous, i.e., we neglectextremity effects and the discrete character of the rods.

We fix the distance R = 2d between the centers of the two rodassemblies and we proceed to calculate the deformation energystored in the membrane (taking into account membranetension). We have, in principle, four degrees of freedom: theheights h1 and h2 of the assemblies with respect to thereference plane that is parallel to the membrane at infinity,and the tilt angle a1 and a2 of the normal to the assembliesrelative to the normal to this reference plane. If we fix thesefour variables we have the following eight boundary conditions:

h01� ¼ a1 � g1; h02� ¼ a2 � g2;

h01þ ¼ a1 þ g1; h02þ ¼ a2 þ g2;

h1� = h1 � a1r, h2� = h2 � a2r,

h1+ = h1 + a1r, h2+ = h2 + a2r. (30)

Here the prime indicates differentiation with respect to x, thesubscript ‘1�’ refers to the position x = �d � r and thesubscript ‘2�’ refers to the position x = d � r. The total energyof the membrane, per unit length and in the limit of smalldeformations, is given by the Helfrich Hamiltonian in theGaussian approximation:51

H ¼ðM

dxk2h002ðxÞ þ g

2h02ðxÞ

h i; (31)

where M = ]�N, �d� r] , [�d + r, d � r] , [d + r, N[. It has tobe minimized with respect to the shape h(x) of the membranein M, but also with respect to h1, h2, a1 and a2, which corre-sponds to requiring that no forces or torques act on the rodassemblies at equilibrium.

Fig. 11 Geometrical parameters for the calculation of the interactionbetween two rod assemblies (side view).

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The corresponding conditions can be obtained directly byperforming two integration by parts on the first variation of H.We obtain

dH ¼ðM

kh0000ðxÞ � gh00ðxÞ½ �dhðxÞdx� G1da1 � G2da2

� F1dh1 � F2dh2; (32)

where we identify G1, G2, F1 and F2 as the torques and forces (perunit length) acting on the rod assemblies, respectively. Requiringthem to vanish we obtain a new set of eight boundary conditions,the conditions that must be satisfied at equilibrium:

G1 k h001þ � h001�� �

� kr h0001þ þ h0001�� �

þ gr h01� þ h01þ� �

¼ 0;

(33)

G2 k h002þ � h002�� �

� kr h0002þ þ h0002�� �

þ gr h02� þ h02þ� �

¼ 0;

(34)

F1 k h0001� � h0001þ� �

þ 2gg1 ¼ 0; (35)

F2 k h0002� � h0002þ� �

þ 2gg2 ¼ 0; (36)

h01þ � h01� � 2g1 ¼ 0; (37)

h02þ � h02� � 2g2 ¼ 0; (38)

h1þ � h1� � r h01� þ h01þ� �

¼ 0 (39)

h2þ � h2� � r h02� þ h02þ� �

¼ 0 (40)

As for the membrane, it must satisfy the Euler–Lagrangeequation:

h00 00(x) � x�2h00(x) = 0. (41)

Solving this linear, one-dimensional problem, yields the inter-action energy per unit length of the rod assemblies

fintðdÞ ¼ 2ffiffiffiffiffikgp

g1g2e�2ðd�rÞ=x; (42)

which, after setting 2r = rrod, 2d = R, 2g1 = Cr1rrod, 2g2 = Cr2rrod

and multiplying by the length of the rod assemblies, yieldseqn (7).

Acknowledgements

This work was supported by JSPS KAKENHI Grant NumberJP25103010 and MEXT as ‘‘Exploratory Challenge on Post-Kcomputer (Frontiers of Basic Science: Challenging the Limits).Numerical calculations were partly carried out on SGI Altix ICEXA at Supercomputer Center of ISSP, University of Tokyo.

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