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in living cells , membrane shape transformation plays a key role in biological functions such as endo / exocytosis and vesicle transports . cell organelles have specific shapes depending on their functions . various types of proteins participate in the regulation of these dynamic and static membrane shapes @xcite . these proteins mainly control local membrane shapes in two ways : hydrophobic insertions ( wedging ) and scaffolding . in the former mechanism , a part of the protein , such as an amphipathic @xmath0-helix , is inserted into the lipid bilayer membrane . in the latter mechanism , the protein domain has a strong affinity for the lipid polar head groups and adsorbs onto the lipid membrane . a bar ( bin / amphiphysin / rvs ) domain , which consists of a banana - shaped dimer , mainly bends the membrane along the domain axis via scaffolding  @xcite . some of the bar superfamily proteins , such as n - bar proteins , also have hydrophobic insertions . experimentally , the membrane tubulation and curvature - sensing by various types of bar superfamily proteins have been observed  @xcite .    objects with rotational symmetry , such as spherical colloids or conical integral proteins inserted perpendicularly to the membrane , generate an isotropic membrane curvature . conversely , bar domains which are banana - shaped generate an anisotropic curvature  @xcite ( amphipathic @xmath0-helices can also yield an anisotropic curvature  @xcite ) . theoretical models have shown that membrane inclusions , such as adsorbed or embedded proteins , or colloids , undergo long - range interactions that are mediated by the curvature elasticity of the membrane  @xcite . it was also shown that anisotropic inclusions experience interactions of longer range than isotropic ones and are able to produce complex aggregates  @xcite . to simplify the theoretical calculations , membrane inclusions are usually modelled as non - deformable objects with a fixed curved shape  @xcite . owing to their small sizes it is often convenient to treat them as point - like objects  @xcite . although the existence of membrane  mediated interactions has been verified experimentally in the case of isotropic inclusions  @xcite , there is no direct experimental evidence yet in the case of anisotropic proteins , despite their biological importance . numerical simulations are therefore essential in this context . atomic and coarse - grained molecular simulations @xcite have been employed to investigate molecular - scale interactions between bar proteins and lipids . the scaffold formation  @xcite and linear assembly  @xcite of bar domains have been demonstrated . to investigate large - scale membrane deformations , a dynamically triangulated membrane model @xcite and meshless membrane models @xcite have been employed ; consequently , various ( meta)stable vesicle shapes  @xcite and the tubule formation dynamics @xcite have been reported . despite these numerous advancements , the assembly of curved protein rods and their membrane shape - remodeling have not been understood so far .    in living cells , more than one type of bar and other proteins cooperatively work to regulate membrane shape . however , in most of the previous reports , rods of a single type are considered in theories and simulations . to our knowledge , only two studies have been reported for the interactions of two types of proteins . the mixture of inclusions with isotropic and anisotropic curvatures was shown to produce the self - assembly of a neckless of anisotropic inclusions around a domain consisting of a lattice of isotropic inclusions , mimicking the assembly of dynamin proteins around a scaffolded bud  @xcite . the phase segregation of rods with different positive spontaneous curvatures was reported in ref .  . in the present study , we investigate the membrane - mediated interactions between two types of protein rods with opposite spontaneous curvatures , theoretically and numerically . this corresponds to the situations in which oppositely curved proteins , such as i - bar and other bar proteins , are adsorbed on the same leaflet of a bilayer membrane , or alternatively , two types of positively 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 possibly different curvatures . we assume that the rods are separated by a distance larger than their size so that we can treat them as point - like anisotropic inclusions using the green function formalism of refs .  . contrary to previous models  @xcite we assume here , in order to model banana - shaped bar domains , that the inclusions fix a curvature in the rod s direction but do not impose any curvature in the orthogonal direction . we also consider the interaction under nonzero surface tension , contrary to what was done in previous works . our results show that rods of same curvature self - assemble into long straight structures that attract rods of opposite curvature on their sides . we then change the scale of our analysis and study the interaction between these macroscopic straight structures . we find that their interaction depends crucially on membrane tension .    in our simulations an implicit - solvent meshless membrane model  @xcite is used to represent a fluid membrane . banana - shaped proteins , assumed to be strongly adsorbed onto the membrane , are modelled together with the membrane region below them as linear strings of particles with a bending stiffness and a preferred curvature . in order to investigate the membrane - mediated interactions , no direct attractive interaction is considered between the rods . we investigate the interaction and structures produced by a mixture of a large number of rods of two types . we find that the results of our simulations agree very well with the theoretical predictions .    in sec . [ sec : theory ] , our multi - scale theoretical analysis of the interactions between curved rods adsorbed onto a membrane with bending rigidity and tension is presented . in sec . [ sec : method ] , the simulation model and method are described . in secs . [ sec : flat ] and  [ sec : ves ] , the assembly of protein rods in flat membranes and vesicles are presented . the summary and discussion are given in sec . [ sec : sum ] . we consider bar - like membrane inclusions shaped as rods of length @xmath1 that are curved in a plane perpendicular to the plane of the membrane . in order to compute the interaction between such inclusions , at separations well larger than @xmath2 , we model them as a point - like inclusions that impose some membrane curvature @xmath3 along the rod direction and no constraint along the orthogonal direction . to derive the interaction between two such rods we use the green function formalism of refs .  . we first discuss the case of a tensionless membrane , then we take into account membrane tension . we first consider two such rods adsorbed onto a membrane with vanishing tension ( @xmath4 ) . let @xmath5 be the vector going from the center of rod  1 to the center of rod  2 , and @xmath6 . we assume that the rods impose curvatures @xmath7 and @xmath8 along the directions @xmath9 and @xmath10 , at angles @xmath11 and @xmath12 with respect @xmath5 and @xmath13 , respectively ( fig . [ fig : geom ] ) . to preserve the symmetry between the two inclusions , we orient @xmath11 counterclockwise and @xmath12 clockwise . we call @xmath14 the distance between the rods . we consider the limit of small membrane deformations . minimizing the helfrich bending energy of the membrane  @xcite with the curvature constraints yields an interaction which is a binary quadratic form in @xmath7 and @xmath8 , as shown in appendix  [ sec : apthe ] . we find that unless @xmath15 , or the opposite , the interaction is well approximated by the sole term proportional to @xmath16 up to distances comparable to @xmath2 . accordingly , in the following we neglect the contributions proportional to @xmath17 and @xmath18 . the curvature - mediated interaction energy between two rods in the absence of membrane tension , @xmath19 , is thus obtained as @xmath20 . \label{eq : hint}\end{aligned}\ ] ] here @xmath21 is the bending rigidity of the membrane . we have exhibited the leading order term @xmath22 , which describes the interaction quantitatively well for @xmath23 larger than a few times @xmath2 . note that this interaction is of longer range than the @xmath24 fluctuation - induced casimir interaction between straight rods and it has a different angular dependence  @xcite . graphs of this interaction are represented in fig . [ fig : th - id]a , b and c for various orientations and curvatures of the rods . when one of the rods is parallel to the separation vector to the other rod ( @xmath25 or @xmath26 ) , eq . ( [ eq : hint ] ) gives @xmath27 which is independent of the orientation of the other rod . in contrast , when the axes of both rods are perpendicular to the separation vector , i.e. , @xmath28 , @xmath29 has the opposite sign and an amplitude three times larger : @xmath30    hence , when two rods are identical , i.e. , @xmath31 , they have a strong attractive interaction at @xmath28 , while they have a weak repulsive interaction at @xmath25 or @xmath26 . obviously , when they are perpendicular to their separation vector they will attract up to contact since they match exactly and produce less deformation when they superimpose . when two rods have curvatures of opposite sign , i.e. , @xmath32 , the interactions are opposite . for @xmath25 or @xmath26 , the rods have a weak attractive interaction . rods of opposite curvatures in a tip - to - tip configuration produce no large - scale curvature deformation when they get close , therefore they will attract up to contact . this will likely be true also for rods not exactly opposite in curvature . thus , for neighboring rods at equilibrium , we expect identical rods to align preferentially side - to - side and rods of opposite curvature to align tip - to - tip . the derivation of the interaction in the presence of membrane tension , @xmath33 , is described in appendix  [ sec : apthe ] , sec . [ tensioncase ] . for the same reasons as previously , we retain in the interaction only the terms proportional to @xmath34 . in the regime of strong tensions , i.e. , for @xmath35 , fig . [ fig : th - id]a , b , and c , show that the interaction @xmath36 displays several similitudes and differences with the tensionless case :    1 .   at short distances the attractive / repulsive behavior is the same as in the tensionless case . hence , identical rods will bind side - to - side and rods of opposite curvature tip - to - tip , as previously . 2 .   the range of the interaction is shorter in the presence of tension : the black curves decay more rapidly that the gray ones . the interaction in the tip - to - tip configuration is not monotonic in the strong tension case ( see fig .  [ fig : th - id]b , black lines ) . since identical rods attract strongly and align side - to - side , while opposite rods align preferentially tip - to - tip , we expect the following scenario . identical rods should aggregate into straight rod assemblies , and parallel rod assemblies should either repel one another , if they are made of rods of same curvature , or attract one another side - to - side if they are made with rods of opposite curvatures . alternate and periodic stripe structures are therefore expected to develop . we detail in appendix  [ sec : apthe2 ] the calculation of the interaction @xmath46 between two coarse - grained parallel rod assemblies of length @xmath47 that are separated by a center - to - center distance @xmath23 . the result is @xmath48 where @xmath49 is the contribution coming from the extremities of the rods , which is subdominant and which we do not evaluate . this interaction confirms that parallel rod assemblies of like curvatures repel while rod assemblies of opposite curvatures attract . the first term , proportional to the length of the rods , vanishes in the absence of tension , i.e. , for @xmath4 . this can easily be understood as flat membrane patches can fit in between the rod assemblies at no cost in the absence of tension . however , some interaction will originate from the extremities of the rod assemblies ( @xmath49 ) , similar to that of isolated rods . in the presence of tension , the energy per unit length will dominate and produce the attraction of rod assemblies of opposite curvature . we see then two properties : ( 1 ) the interaction is short - ranged , decaying over @xmath37 , ( 2 ) the larger the tension the stronger the interaction . since the details of the meshless membrane model and protein rods are described in ref .   and refs . , respectively , we briefly describe the model here . a fluid membrane is represented by a self - assembled one - layer sheet of @xmath50 particles . the position and orientational vectors of the @xmath51-th particle are @xmath52 and @xmath53 , respectively . the membrane particles interact with each other via a potential @xmath54 . the potential @xmath55 is an excluded volume interaction with a diameter @xmath56 for all pairs of particles . solvent is implicitly accounted for by an effective attractive potential @xmath57 . the bending and tilt potentials are given by @xmath58 and @xmath59 w_{\rm { cv}}(r_{i , j})$ ] , respectively , where @xmath60 , @xmath61 , @xmath62 , @xmath63 is a weight function , and @xmath64 denotes the thermal energy . the spontaneous curvature @xmath65 of the membrane is given by @xmath66 . @xcite in this study , @xmath67 and @xmath68 except for the membrane particles belonging to the protein rods . a protein rod is modeled as a linear chain of @xmath69 membrane particles . we use @xmath70 , which corresponds to the typical aspect ratio of the bar domains . the bar domain width is approximately @xmath71 nm , and the length ranges from @xmath72 to @xmath73 nm . @xcite two types of protein rods , called rods 1 and 2 , are used . as detailed below , rods 1 and 2 have positive and negative spontaneous curvatures @xmath74 and @xmath75 along the rod axis , respectively , and both rods have no spontaneous curvature perpendicular to the rod axis . hereafter , we call the membrane particles forming a protein rod _ protein particles_. the protein particles in each protein rod are further connected by a bond potential @xmath76 where @xmath77 , @xmath78 , and @xmath79 are used . the bending potential is given by @xmath80 , where @xmath81 and @xmath82 for the rod 1 and 2 , respectively . for bonded pairs of protein particles , we use four times larger values of @xmath83 and @xmath84 , together with the corresponding spontaneous curvature , to prevent the rod from bending tangentially in the membrane . we employ the parameter sets used in ref .  . the membrane has mechanical properties that are typical for lipid membranes : bending rigidity @xmath85 , area of the tensionless membrane per particle @xmath86 , area compression modulus @xmath87 , and edge line tension @xmath88 . this edge tension @xmath89 is sufficiently large to prevent membrane rupture in this study  @xcite . molecular dynamics with a langevin thermostat is employed  @xcite . in the following , the results are displayed with the rod length @xmath90 for the length unit , @xmath64 for the energy unit , and @xmath91 for the time unit , where @xmath92 is the diffusion coefficient of the membrane particles in the tensionless membrane  @xcite . for the rod assembly on a flat membrane , the @xmath93 ensemble with periodic boundary conditions is used . the projected area @xmath94 is fluctuated for a constant surface tension @xmath95 while maintaining the aspect ratio @xmath96  @xcite . the rod-1 curvature is fixed as @xmath97 , which can induce membrane tubulation with a circumference of @xmath98 for the present bending elastic constants . flat membranes with @xmath99 are investigated with various values of @xmath100 . the density of the rods 1 and 2 are set to @xmath101 and @xmath102 , where @xmath103 and @xmath104 are the numbers of the rods 1 and 2 , respectively .    for the rod assembly on a vesicle , the @xmath105 ensemble is used at @xmath106 and @xmath107 . the radius of the vesicle is @xmath108 in the absence of the rods . for the annealing simulations , the rod curvatures are changed from @xmath109 to @xmath97 while maintaining the ratio @xmath100 with annealing rate @xmath110 and subsequently the vesicle is equilibrated for @xmath111 . for each simulation condition , the reproducibility is confirmed at least from four different initial conformations . ( a ) the ratio of the rod curvatures , @xmath112 . ( b ) @xmath113 . ( c ) @xmath114 . ( d ) @xmath115 . the red and blue particles represent the segments of the rod 1 and 2 , respectively . the gray particles represent membrane particles . ]     and @xmath4 . ] first , we describe the rod assemblies on the tensionless flat membrane ( @xmath4 ) . figure  [ fig : snap_cs ] shows typical snapshots . for zero rod - curvature of the rods 2 ( @xmath116 ) , the rods 1 form disk - shaped tubules like in the absence of rods 2 . the rods 2 are distributed on the flat region of the membrane . small tubules have a semicircular - disk shape . in contract , large tubules have a disk shape ( like a mussel shell ) connected to the flat membrane by a narrow cylindrical neck ( see the tubules in fig .  [ fig : snap_cs]a ) . this assembly of the rods 1 agrees with the theoretical prediction for the point - like inclusions in sec . [ sec : tpoint ] ( identical curved rods attract and align side - to - side ) . identical rods 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 band curved along the short axis of the assembly ( parallel to the rod axis ) . eventually , long band assemblies bend along their long axis and form a tubule , as this reduces the bending energy cost around the assembly . the rods 2 , with @xmath117 , exhibit little interactions with the rods 1 . this also agrees with the prediction of eq . ( [ eq : hint ] ) .    as @xmath118 increases , the necks of the large tubules remain open . for @xmath113 , the rod-1 assemblies exhibit wide walls or hill shapes ( see fig . [ fig : snap_cs]b ) . the rods 2 are concentrated around the foot of these walls and stabilize their negative curvature . with further increasing @xmath118 , the rods 2 more clearly assemble at both sides of the rod-1 assemblies and form long straight bumps [ see fig .  [ fig : snap_cs]c and the corresponding movie provided in esi ( movie 1 ) ] . these structures recall the shape of a centipede or myriapod : the rod-1 straight assembly looks like the body of a centipede and the rods 2 on both sides are like many legs . this attraction of oppositely curved rods along their axis also agrees with the theoretical prediction for the point - like inclusions in sec . [ sec : tpoint ] ( oppositely curved rods attract and align tip - to - tip ) . the straight assemblies are also stable at larger @xmath118 . however , at @xmath119 , straight assemblies of opposite curvature have attractive interaction in the lateral direction ( parallel to the axis of each rod ) and form a periodic bump structure ( see fig . [ fig : snap_cs]d ) . each rod-1 ( -2 ) straight assembly is connected to two rod-2 ( -1 ) straight assemblies on both lateral sides . we call this assembly of the rods 1 and 2 a _ stripe assembly _ , since it forms periodic bands . this stripe assembly is metastable but has a longer life time than our simulation periods at @xmath115 . even when the stripe assembly is set as an initial conformation , the rod-1 straight assemblies separate to form isolated bumps at @xmath120 .     and ( b ) mean vertical rod span @xmath121 for @xmath114 , @xmath122 , and @xmath123 at @xmath124 and @xmath125 . at @xmath126 and @xmath4 , the same data are used as in fig . [ fig : snap_cn5 ] . ]    , @xmath122 , and @xmath123 at @xmath4 . ( b ) @xmath124 , @xmath127 , @xmath125 , @xmath128 , and @xmath129 at @xmath114 . the same data are used as in fig . [ fig : ar ] for the corresponding parameter sets . ]     for ( a ) @xmath130 and ( b ) @xmath131 . ]     and the ratio of the rod curvatures @xmath118 . ]    figure  [ fig : snap_cn5 ] shows the assembly dynamics at @xmath126 [ the corresponding movie is provided in esi ( movie 2 ) ] . first , short straight assemblies are formed . as the tips of the bumps approach closely , they fuse into one large straight assembly . in contrast , as they approach laterally , they form stripe assembly of short straight assemblies . as a long bump approaches , the stripe assemblies are disassembled and reassemble into longer assemblies ( see the upper region of the three right snapshots in fig .  [ fig : snap_cn5 ] ) . note that some rods of type 2 also contact the rod-1 assemblies at the tips of the assemblies ( see snapshots at @xmath132 and @xmath133 in fig . [ fig : snap_cn5 ] ) , so that an attractive interaction exists between the rods of types 1 and 2 even when the rod-1 axis is perpendicular to the separation vector . this attraction is in good agreement with the prediction of eq . ( [ eq : hint ] ) with @xmath25 and @xmath134 . as the rods assemble , the projected membrane area @xmath135 decreases while the mean vertical rod span @xmath121 and the mean cluster size @xmath136 of rods 1 increase ( see figs .  [ fig : ar ] and [ fig : cl ] ) . at @xmath137 , the rod-1 cluster formation becomes slower than at @xmath138 , since the stripe assembly suppresses the fusion between the rod-1 assemblies . the vertical span is calculated from the height variance of all rod segments as @xmath139 where @xmath140 . a rod is considered to belong to a cluster when the distance between the centers of mass of the rod and one of the rods in the cluster is less than @xmath141 . for the rods 1 , the mean cluster size is given by @xmath142 with @xmath143 where @xmath144 is the number of clusters of size @xmath145 .    at small positive surface tensions ( @xmath146 , _ i.e. _ , @xmath147 ) , the bending energy contribution should be dominant in the interactions between the rods as described in sec . [ sec : theory ] . our simulation results support this theoretical prediction . the rod assembly is only slightly modified at @xmath148 ( @xmath149 ) . in contrast , a large positive surface tension ( @xmath150 ) suppresses both the decrease of the projected area @xmath135 and the protrusion of the rod assemblies ( see fig .  [ fig : ar ] ) . since the curvature - energy gain by the rod assembling is reduced when increasing @xmath95 ( see the theory in sec . [ sec : theory_tension ] and fig . [ fig : th - id ] ) , the rod cluster size decreases and more rods remain in an isolated state ( see figs . [ fig : cl]b and [ fig : snap_f ] ) . interestingly , the stripe assembly is stabilized for smaller @xmath118 at larger values of @xmath95 , as shown in fig . [ fig : pd ] . for @xmath114 , the stripe assembly exists for @xmath151 while it does not for @xmath152 ( compare figs .  [ fig : snap_cs]c and [ fig : snap_f ] ) . this agrees well with the theoretical results of sec . [ sec : tbars ] showing that the attraction between long straight assemblies of opposite curvature scales as @xmath153 and thus increases with membrane tension . of the rods spontaneous curvatures . ( a)(d ) snapshots . the membrane particles are represented by a transparent grey 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 orientation vector @xmath53 lies along the direction from the yellow ( green ) to red ( blue ) hemispheres . ( a ) , ( b ) vesicles formed by annealing to @xmath97 with ( a ) @xmath154 and ( b ) @xmath155 . ( c ) vesicles formed by an increase in @xmath118 from @xmath156 to @xmath123 at @xmath97 . ( d ) vesicles formed by a decrease in @xmath118 from @xmath157 to @xmath158 at @xmath97 . top and bottom panels show the snapshot of all particles in birds - eye view and a sliced snapshot in front view , respectively . ( e),(f ) dependence of ( e ) the mean radius of gyration @xmath159 and ( f ) mean asphericity @xmath160 of vesicles on @xmath118 . the data labeled up and down are obtained by an increase and decrease in @xmath118 at @xmath97 . , title="fig : " ]   of the rods spontaneous curvatures . ( a)(d ) snapshots . the membrane particles are represented by a transparent grey 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 orientation vector @xmath53 lies along the direction from the yellow ( green ) to red ( blue ) hemispheres . ( a ) , ( b ) vesicles formed by annealing to @xmath97 with ( a ) @xmath154 and ( b ) @xmath155 . ( c ) vesicles formed by an increase in @xmath118 from @xmath156 to @xmath123 at @xmath97 . ( d ) vesicles formed by a decrease in @xmath118 from @xmath157 to @xmath158 at @xmath97 . top and bottom panels show the snapshot of all particles in birds - eye view and a sliced snapshot in front view , respectively . ( e),(f ) dependence of ( e ) the mean radius of gyration @xmath159 and ( f ) mean asphericity @xmath160 of vesicles on @xmath118 . the data labeled up and down are obtained by an increase and decrease in @xmath118 at @xmath97 . , title="fig:",width=302 ] the rods also assemble on vesicles like on flat membranes . however , the assembly structures are largely modified by the original vesicle curvature and its closed geometry . when a single type of rod exists on the vesicle , the rods induce oblate or polyhedral shapes of the vesicle and they assemble at the edges of the polyhedron  @xcite . for negative rod curvatures , invaginations into the inside of the vesicle can occur  @xcite . here , we set the rod-1 density , @xmath161 , to make the vesicle form a disk shape in the absence of the rods 2 . we add the rods 2 with the same density and investigate how the vesicle shape is changed upon varying @xmath118 . figure  [ fig : snap_ves]a and b show the snapshots obtained by annealing from @xmath109 to @xmath97 at fixed ratio @xmath118 . for @xmath154 , the vesicles form a disk shape with a concave circular region between the center and the rim ; the rods 2 are aligned along radial directions of the disk in this concave region . for @xmath155 , the rods 2 form invaginations into the inside of the vesicle and the two rims of the rod-1 assembly are partially connected by the rod-2 assembly like in the flat membrane . for intermediate values of @xmath118 , the obtained vesicle shapes depend on the initial conformations and thermal noise during the annealing : some vesicles have an invagination or partial doubled rims .    to investigate the stability of the disk - shaped and invaginated vesicles , we simulated vesicles with different @xmath118 starting from the annealed conformation at @xmath154 and @xmath157 as initial states . as @xmath118 increases for disk - shaped vesicles , the center region of the vesicle becomes more spherical . this shape change reduces the radius of the disk , resulting in a buckling or winding of the disk rim , at @xmath126 and @xmath123 ( see fig . [ fig : snap_ves]c ) . at @xmath162 and @xmath157 , the rods 2 form a straight dimple on the side of the rod-1 assembly ( as seen in the left region of the sliced snapshot in fig . [ fig : snap_ves]b ) as well as an invagination . starting from an invaginated vesicle , the invagination swells to an ellipsoidal shape and the rest of the rods 1 form a circular rim , at @xmath114 ( see fig .  [ fig : snap_ves]d ) . at @xmath154 , the invagination is removed and a disk - shaped vesicle is obtained [ see the movie is provided in esi ( movie 3 ) ] . these shape changes can be characterized by the radius of gyration @xmath163 and the shape parameter , asphericity @xmath164 ( see fig .  [ fig : snap_ves]e and f ) . the asphericity is the degree of deviation from a spherical shape and is expressed as  @xcite @xmath165 where @xmath166 and @xmath167 are the eigenvalues of the gyration tensor of the vesicle : @xmath168 , where @xmath169 and @xmath170 are the coordinates of the center of mass . for a perfect sphere , @xmath171 , for a thin rod @xmath172 , and for a thin disk , @xmath173 . starting from a disk - shaped vesicle ( labeled ` up ' in fig . [ fig : snap_ves]e and f ) , the vesicles have more spherical and less compact shapes than invaginated vesicles so that they take larger @xmath163 and smaller @xmath164 . thus , the vesicle shapes can be chosen from many metastable conformations by using the hysteresis of @xmath118 variations . our simulations demonstrate a control method of the vesicle shapes by using hysteresis . we have studied the assembly structures of binary banana - shaped protein rods . compared to the situation where only one type of rod is present , the coexistence of two types of rods , with opposite curvatures , induces a greater variety of membrane shapes . here , using meshless membrane simulations , we demonstrated straight bumps and stripe structures in flat membranes , and buckled rims and ellipsoidal invaginations in vesicles . more complicated shapes such as vesicles with periodic bumps , like cartridge pleats , can be formed for larger vesicles . these structure formations can be examined in experiments with two types of bar proteins absorbed on the same side of the membrane or with a single type of protein absorbed on both side of the membrane . the periodic length of the stripe structure is the sum of the longitudinal lengths of two bar domains ( @xmath174 to @xmath125 nm ) . it should therefore be observable by electron microscopy .    for rods of like curvatures , our theory predicts a side - to - side attractive interaction and a tip - to - tip repulsion . for rods having curvatures 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 in the side - to - side configuration and thus build straight rod assemblies ; these assemblies attract the rods with opposite curvatures in the tip - to - tip configuration thereby inducing a side - to - side alignment of the formed assemblies . the formation of those alternate assemblies , or stripe structures , therefore agrees well with our theoretical predictions .    with increasing membrane tension , we theoretically found that the attractive interaction between opposite rods in the tip - to - tip configuration ( which corresponds to the orientations involved in the stripe structures ) is increased at short distances and decreased at large distances ( see fig . [ fig : th - id]b ) . likewise , our larger - scale analysis of sec . [ sec : tbars ] predicts a short - range attraction between opposite straight rod assemblies that increases with membrane tension . these theoretical results explain well the observed increased stability of the stripe structures when the tension is increased and the shorter length of the rod assemblies in the thermalized system . these theoretical predictions are supported by our simulation results although the theory considers rigid rods with a fixed shape while the rods in the simulation are flexible . the real proteins are not completely rigid but their stiffness is not measured . the rods flexibility can be considered to result in an effectively smaller rod curvature , not altering the sign of the interactions in the present assemblies . the protein stiffness , together with the finite size of the rod inclusions , should however be taken into account for quantitative predictions of the assembly conditions . the effects of the casimir force , induced by the fluctuations of the membrane , remains to be investigated further . it was ignored in the present work since it is normally sub - dominant for small inclusions imposing large membrane deformations . this casimir force is known however to induce an effective attraction between long straight rods  @xcite . we have investigated straight rods in our simulations , and found that the latter have too weak casimir interactions to induce rod clusters for the present length ( @xmath70 ) . however , twice longer rods ( @xmath175 ) can form clusters . thus , the assembly of longer rods can be induced by both casimir and rod - curvature forces .    in living cells , many types of bar superfamily proteins cooperate to regulate membrane shapes . here , we have studied only the mixture of two types of proteins . the cooperative effect of mixing more than two proteins is an important topic for further studies . this work was supported by jsps kakenhi grant number jp25103010 . in the small deformation regime , the membrane shape can be expressed in the monge gauge by a height function @xmath176 , with @xmath177 a point of the projected plane . the helfrich elastic energy of the membrane is given , to quadratic order , by  @xcite @xmath178 = \frac{\kappa}{2}\int\!d^2r\,h\mathcal{l}h,\end{aligned}\ ] ] with @xmath179 , the operator associated with the hamiltonian , and @xmath180 the coherence length arising from the membrane tension @xmath95 and the membrane bending rigidity @xmath21 . let us consider a point - like inclusion , placed at @xmath181 , that imposes a membrane curvature @xmath182 along the direction given by the unit vector @xmath9 of angle @xmath11 ( see the definition of the angles in sec . [ sec : tpoint ] ) , and a second point - like inclusion , placed at @xmath183 , imposing a curvature @xmath184 along the direction given by the unit vector @xmath10 of angle @xmath12 . we shall discuss later on how to mathematically fix the  size " of these inclusions . the constraints set by the inclusions , for small membrane deformations , read @xmath185 where a comma indicates differentiation . here @xmath186 is the polar angle of the direction @xmath10 with respect to the @xmath187-axis , i.e. , @xmath188 . minimizing the helfrich energy  ( [ eq : hel ] ) with the above constraints yields the euler  lagrange equation : @xmath189 + \lambda_2\big [ \delta_{,xx}(\bm{r}-r\bm{e}_x)\cos^2\theta'_2   \nonumber \\ & + \delta_{,xy}(\bm{r}-r\bm{e}_x)\sin(2\theta'_2 ) + \delta_{,yy}(\bm{r}-r\bm{e}_x)\sin^2\theta'_2 \big ] . \nonumber\end{aligned}\ ] ] here , @xmath190 is the dirac distribution , and @xmath191 and @xmath192 are lagrange multipliers . the solution is given by @xmath193 + \lambda_2\big [ g_{,xx}(\bm{r}-r\bm{e}_x)\cos^2\theta'_2   \nonumber \\ & + g_{,xy}(\bm{r}-r\bm{e}_x)\sin(2\theta'_2 ) + g_{,yy}(\bm{r}-r\bm{e}_x)\sin^2\theta'_2 \big ] , \nonumber\end{aligned}\ ] ] where @xmath194 is the green function defined by @xmath195      if @xmath4 , the euler - lagrange operator becomes @xmath196 and the green function is given by  @xcite @xmath197 indeed , with @xmath198 ( cylindrical coordinates ) , the most general rotationally symmetric solution of @xmath199 is @xmath200 . the finiteness of @xmath201 at the origin requires @xmath202 and integrating eq . ( [ eq : green ] ) around @xmath203 yields @xmath204 . the constants @xmath205 and @xmath206 are useful if one wishes to satisfy some boundary condition for large @xmath207 , however they do not contribute to the fourth derivatives of @xmath201 appearing below , so we can set @xmath208 , which yields eq . ( [ eq : freegreen ] ) .    satisfying the constraints ( [ cons1 ] ) and ( [ cons2 ] ) gives a linear set of equations for the lagrange multipliers : @xmath209 where @xmath210 , @xmath211 and @xmath212 is a matrix such that @xmath213 \nonumber\\ & + 2g_{,xyyy}(r\bm{e}_x)\sin\theta_1\sin\theta'_2\sin(\theta_1+\theta'_2 ) \nonumber\\ & + g_{,yyyy}(r\bm{e}_x)\sin^2\theta_1\sin^2\theta'_2\,,\end{aligned}\ ] ] and with @xmath214 obtained from @xmath215 by replacing @xmath216 by @xmath217 , and with @xmath218 obtained from @xmath219 by replacing @xmath11 by @xmath186 . using eqs . ( [ eq : hel ] ) and  ( [ eq : lh ] ) , one obtains exactly @xmath220 we need the fourth derivatives of the green function . from eq . ( [ eq : freegreen ] ) , we obtain @xmath221 it follows that the fourth derivatives of @xmath222 , that appear in @xmath219 and @xmath218 , are singular . this problem comes from the fact that we use mathematically a point - like constraint . physically , the inclusions have a characteristic size , here @xmath223 , and we can specify this size by using an upper wavevector cutoff @xmath224 for the membrane deformation in the reciprocal space . we take numerically @xmath225 as in refs .  , since this choice was shown to match the exact calculations in the case of isotropic inclusions . this gives @xmath226 similarly , @xmath227 we thus obtain @xmath228 where we have switched back to @xmath229 . note that @xmath214 is symmetric upon the exchange of @xmath11 and @xmath12 as it should be . using eq . ( [ hbyparts ] ) , we obtain exactly @xmath230 where @xmath231 . however , since the rods we are actually modeling are not point - like , the above formula is not meaningful for @xmath232 . we therefore take the leading order of @xmath233 in @xmath234 , which describes the interaction quantitatively well for @xmath23 larger than a few times @xmath235 . replacing @xmath235 by @xmath2 yields the interaction @xmath236 given by eq . ( [ eq : hint ] ) . note that this result is symmetric upon exchanging @xmath11 and @xmath12 and invariant in changing @xmath11 into @xmath237 or @xmath12 into @xmath238 , as it should be .      in the presence of membrane tension , for @xmath239 , the formalism in the previous section is still valid , provided the full green function is used , i.e. , the green function of @xmath240 defined by eq . ( [ eq : green ] ) . in particular , eqs . ( [ eq : m11])([hbyparts ] ) still hold . the green function in the presence of tension is given by  @xcite @xmath241,\ ] ] where @xmath242 and @xmath243 ( used below ) are modified bessel functions . indeed , the most general rotationally symmetric solution of @xmath244 is @xmath245 . the finiteness of @xmath201 at the origin requires @xmath246 and integrating eq . ( [ eq : green ] ) around @xmath203 yields @xmath247 . discarding the constant term @xmath205 and the term @xmath248 that diverges at infinity , we obtain eq . ( [ eq : tensiongreen ] ) . note that eq . ( [ eq : tensiongreen ] ) reduces to eq . ( [ eq : freegreen ] ) for @xmath249 .    regularizing the fourth derivatives of the green function as previously , we obtain @xmath250 similarly , @xmath251 using eqs . ( [ eq : m11])([hbyparts ] ) with these elements , we obtain the graphs of @xmath33 shown in sec . [ sec : theory_tension ] and the corresponding asymptotic behaviors @xmath43 and @xmath45 . we consider two parallel rod assemblies , one made of rods of curvature @xmath7 and the other one made of rods of curvature @xmath8 . recall that the rods are curved in the direction perpendicular to the axes of the rod assemblies . let us define @xmath207 , @xmath252 and @xmath253 in such a way that the curvatures @xmath7 and @xmath8 corresponds to angular variations @xmath254 and @xmath255 over the distance @xmath256 , respectively ( see fig . [ fig : assemblies ] ) . for the sake of simplicity , we regard the rod assemblies as infinite and homogeneous , i.e. , we neglect extremity effects and the discrete character of the rods . we fix the distance @xmath257 between the centers of the two rod assemblies and we proceed to calculate the deformation energy stored in the membrane ( taking into account membrane tension ) . we have , in principle , four degrees of freedom : the heights @xmath258 and @xmath259 of the assemblies with respect to the reference plane that is parallel to the membrane at infinity , and the tilt angle @xmath260 and @xmath261 of the normal to the assemblies relative to the normal to this reference plane . if we fix these four variables we have the following eight boundary conditions : @xmath262 here the prime indicates differentiation with respect to @xmath187 , the subscript ` @xmath263 ' refers to the position @xmath264 and the subscript ` @xmath265 ' refers to the position @xmath266 . the total energy of the membrane , per unit length and in the limit of small deformations , is given by the helfrich hamiltonian in the gaussian approximation  @xcite : @xmath267,\end{aligned}\ ] ] where @xmath268-\infty ,- d - r]\cup[-d+r , d - r]\cup[d+r,\infty[$ ] . it has to be minimized with respect to the shape @xmath269 of the membrane in @xmath270 , but also with respect to @xmath258 , @xmath259 , @xmath260 and @xmath261 , which corresponds to requiring that no forces nor torques act on the rod assemblies at equilibrium . the corresponding conditions can be obtained directly by performing two integration by parts on the first variation of @xmath271 . we obtain @xmath272\delta h(x)\,dx \nonumber\\ & -\gamma_1\,\delta\alpha_1-\gamma_2\,\delta\alpha_2 -f_1\,\delta h_1-f_2\,\delta h_2,\end{aligned}\ ] ] where we identify @xmath273 , @xmath274 , @xmath275 and @xmath276 as the torques and forces ( per unit length ) acting on the rod assemblies , respectively . requiring them to vanish we obtain a new set of eight boundary conditions , the conditions that must be satisfied at equilibrium : @xmath277 as for the membrane , it must satisfy the euler - lagrange equation @xmath278    solving this linear , one - dimensional problem , yields the interaction energy per unit length of the rod assemblies @xmath279 which , after setting @xmath256 , @xmath280 , @xmath281 , @xmath282 and multiplying by the length of the rod assemblies , yields eq . ( [ intbars ] ) .
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 , both for zero surface tension and for finite surface tension . on a larger scale , 
 the interactions between 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 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 stripe structure . 
 positive surface tension is found to stabilize the stripe formation . 
 the simulation results agree well with the theoretical predictions .