Curvature variation controls particle aggregation on fluid vesicles

Afshin Vahid1, Anđela Šarić2, Timon Idema1∗1Department of Bionanoscience, Kavli Institute of Nanoscience,

Delft University of Technology, Delft, The Netherlands and2 Department of Physics and Astronomy, Institute for the Physics of Living Systems,

University College London,London, United Kingdom(Dated: September 14, 2018)

Cellular membranes exhibit a large variety of shapes, strongly coupled to their function. Manybiological processes involve dynamic reshaping of membranes, usually mediated by proteins. Thisinteraction works both ways: while proteins influence the membrane shape, the membrane shapeaffects the interactions between the proteins. To study these membrane-mediated interactions onclosed and anisotropically curved membranes, we use colloids adhered to ellipsoidal membrane vesi-cles as a model system. We find that two particles on a closed system always attract each other,and tend to align with the direction of largest curvature. Multiple particles form arcs, or, at largeenough numbers, a complete ring surrounding the vesicle in its equatorial plane. The resulting vesi-cle shape resembles a snowman. Our results indicate that these physical interactions on membraneswith anisotropic shapes can be exploited by cells to drive macromolecules to preferred regions ofcellular or intracellular membranes, and utilized to initiate dynamic processes such as cell division.The same principle could be used to find the midplane of an artificial vesicle, as a first step towardsdividing it into two equal parts.

INTRODUCTION

Cellular membranes are two-dimensional fluid inter-faces that consist of a large variety of components. Theyform the boundary between the cell and the outsideworld, and, for eukaryotic cells, separate the inside of thecell into numerous compartments known as organelles. Inorder for biological processes like cell division, vesiculartrafficking and endo/exocytosis to occur, cellular mem-branes have to reshape constantly. Consequently, mem-branes exhibit a variety of morphologies, from a sim-ple spherical liposome to bewildering complex structureslike interconnected tubular networks as found in Mito-chondria and the Endoplasmic Reticulum (ER), or con-nected stacks of perforated membrane sheets in the Golgiapparatus1–4. There are different mechanisms by whichmembranes achieve these structures, the most impor-tant of which is through the interplay between membranelipids and various proteins[5–7]. A biological membraneis home to different types of proteins that are adheredto or embedded in it. These proteins deform the mem-brane and, consequently, they can either repel or attracteach other [8–13]. Spatial organization of such proteinsin biological membranes is essential for stabilizing themembrane and for the dynamic behaviour of cellular or-ganelles [14–17].

Recently, it has been experimentally [18, 19] and theo-retically [8, 9, 11, 20–25] revealed that membrane-curvingparticles, like colloids or identical proteins, adhered to amembrane self-assemble into striking patterns. For in-stance it has been shown that colloids adhered to a spher-ical membrane form linear aggregations [23]. In all of thestudies to date, the global shape of the membrane is se-lected from one of three options: planar, spherical, ortubular. These global membrane shapes impose a ho-

mogeneous background curvature, which is consideredto be conserved throughout the process under investiga-tion. Outside factors changing the membrane have notyet been included in the study of membrane-mediatedinteractions. Membranes in cellular compartments suchas in ER and the Golgi complex are however dynamicentities and possess peculiar shapes forming regions withhigh local curvature and regions with less curvature [3].Forming and stabilizing such shape inhomogeneities isnecessary for cellular functions like sensing and traffick-ing [26]. It is therefore warranted to investigate how theinteractions between membrane inclusions are affected byanisotropies in the membrane curvature.

In this study, through a numerical experiment, we in-vestigate the interactions between colloids adhered to aquasi-ellipsoidal membrane with a varying curvature. Wealso include all other factors from earlier studies suchas surface tension, adhesion energy (required for colloidsto adhere to the membrane) and constant volume ef-fects. We use a dynamic triangulation network to modelthe membrane, and computationally minimize the totalenergy of the membrane via a Monte Carlo algorithm.Firstly, we show that the interaction between two colloidsadhered to spherical vesicles is significantly affected bythe vesicle curvature. Secondly, we demonstrate that lin-ear aggregates of colloids exploit the curvature anisotropyand adjust their orientation to minimize the total en-ergy on a quasi-ellipsoidal membrane. Using umbrellasampling, we further show that the total energy of themembrane favors two colloids to attract each other atthe mid-plane of a prolate ellipsoid that is perpendicularto its major axis. Finally, we investigate how the vari-ous terms in the total energy of the membrane affect thestrength of the interactions. Our results show that thevariation in the membrane shape can play a crucial role

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SPHERICAL MEMBRANE

FIG. 1. The curvature energy of the membrane for spherical vesicles of sizes Dv = 14σ (circles) and Dv = 19.5σ (pluses)containing two colloids. The force between colloids in the smaller vesicle is stronger and has a larger interaction range.

in a variety of cellular functions that require macromolec-ular assembly or membrane remodeling.

MODEL

The conformation of a fluid membrane can be de-scribed as the shape minimizing the classical Helfrichenergy functional [27]:

uCurv =κ

2

∫A

(2H)2dA, (1)

where H is the mean curvature at any point on thesurface of the membrane and geometrically is definedas the divergence of the normal vector to the surface,H = − 1

2∇ · n. In our computational scheme, we dis-cretize the membrane by a triangulated network, whosetriangles represent course-grained patches of the mem-brane [28, 29]. Using a discretized form of the Helfrichenergy, we define the curvature energy as:

uCurv = κ∑<ij>

1− ni · nj , (2)

where ni and nj are the normal vectors to any pair ofadjacent triangles i and j, respectively. The summationruns over all pairs of such triangles. In order to guar-antee the fluidity of the membrane, we cut and reattachthe connection between the four vertices (which we la-bel with and refer to as beads) of any two neighboringtriangles. The membrane in our system does not un-dergo any topological changes and we can thus ignorethe Gaussian curvature contribution in the bending en-ergy. We impose the conservation of membrane surfacearea (A) and enclosed volume (V ) by adding the terms

uA = KA(A−At)2/At and uV = KV(V − Vt)2/Vt to the

energy during the minimization process, with At and Vtthe target values of the membrane’s area and enclosedvolume. The corresponding constants are chosen suchthat both the area and volume deviate less than 0.05%from their target values. To enable colloids to adhereto the membrane, we introduce an adhesion potential,uAd = −ε(lm/r)6, between colloids and the membrane,where ε is the strength of the adhesion energy and, r andlm are, respectively, the center to center distance and theminimum allowed separation between colloids and mem-brane beads. Finally, we need to give the membranean anisotropic shape for which we deform our sphericalmembrane into a prolate ellipsoid. In order to do so, weintroduce two weak (compared to the strength of the ad-hesion energy) spring-like potentials between two smallareas of the vesicle (the two poles of the ellipsoid) and thecenter of the vesicle, uEll = KEll(L − a)2; KEll, a and Lare the potential strength, the major axis of the ellipsoidand the length of any line connecting the beads situatedat the poles of the ellipsoid to the center, respectively.Since the adhesion energy is stronger than the appliedharmonic potential, colloids effectively do not feel anydifference between the energy cost for bending the mem-brane at these two areas and at the regions belonging tothe rest of the ellipsoid. We verified this claim by con-sidering a spherical membrane, and find that there is nosignificant difference between the case of including uEll

with a being the radius of the vesicle, and the case we donot include such a potential.Having defined all the contributions to the total energyof the membrane (uTotal = uCurv +uA +uV +uAd +uEll),we perform Monte Carlo simulations to reach the equilib-rium shape of an ellipsoid containing an arbitrary numberof colloids. To do so, we implement the Metropolis al-gorithm, in which we have three types of moves: we can

3

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16

5 6 7 8 9 10 11

e = 1.351

e = 1.411

e =1.503

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-10

-5

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5

0

e = 1.503

e = 1.411

e = 1.351

0

4

8

12

16

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e =1.411

e=1.501

/k T

B

D ( ) D ( )

/8 2 /8 3 /8

a) b)

c) d) i. ii.

iii.

e = 1.351

/2

5

D

D/k

T

B

/k T

B

FIG. 2. Colloids adhered to a quasi-ellipsoidal membrane behave differently in different directions. Decreasing the ellipticityof the vesicle, which is defined as e = a/b, (a) strengthens the attractions between the colloids along the major axis and(b) weakens the interaction between the colloids along the minor axis. Figure (c) illustrates that the energy of a membranecontaining a pair of colloids decreases when the angle between the pair and the semi-major axis increases.

modify the position of a random bead of the membrane,impose a rearrangement in the connections of beads, ormove the colloids around. The first two moves are ener-getically evaluated based on the total energy, while anychanges in the position of colloids are only based on theadhesion energy.During the simulations, we keep the number of par-ticles constant and set all the relevant parameters as:κ = 36kBT , ε = 8.5kBT , KA = 2 × 103kBT/σ

2, KV =250kBT/σ

3 and KEll = 0.1kBT/σ2, where kBT is the

thermal energy and σ is the diameter of the beads con-structing the membrane. The diameter of the colloids isset to σColl = 5σ.

RESULTS AND DISCUSSION

First, we analyze the interaction between two colloidsadhered to the surface of two vesicles of different sizes.We keep the size of colloids and beads the same in bothcases. We use umbrella sampling [30] to calculate the ex-cess energy of the membrane as a function of the distancebetween the colloids. In effect, we apply a harmonic po-tential u = 1

2k(D−D0)2, as our biased potential, betweenthe two colloids directed along the coordinate of interestin order to restrain the system to sample around eachdistance D0. Having performed the sampling process, weuse the weighted histogram analysis method (WHAM)for obtaining the optimal estimate of the unbiased prob-ability distribution, from which we can calculate the freeenergy of the system. The free energy is calculated

with respect to the initial position of the colloids ∆E =E(at the coordinate of interest) − E(initial coordinate).The excess area of the membrane available for colloidsto adhere to is equal in both vesicles. As illustrated inFig. 1, the depth of the excess energy of the membranewith a smaller radius is significantly larger. In contrast,for the larger vesicle after a short distance colloids donot feel each other and the energy becomes flat. As theonly difference between two test cases is the curvature,we conclude that this effect is due to vesicles being ofdifferent radii.

Next, we examine the interaction between two colloidson the surface of a quasi-ellipsoidal membrane. We po-sition the colloids symmetrically along the major axis ofthe ellipsoid (see Fig. 2d(i) ). We repeat the samplingprocedure for different aspect ratios, e = a/b, of the el-lipsoid. Since the volume is conserved during the shapeevolution, one can easily calculate the semi-minor axis, b,as: b =

√3V/4πa. As depicted in Fig. 2a, along the ma-

jor axis colloids attract each other in order to minimizeboth the adhesion and curvature energies. Decreasingthe asphericity of the ellipsoid (e → 1.0+) in this direc-tion enhances the deformable area (i.e. the number ofaccessible beads), hence the strength of the attractionenergy increases. Similarly, particles that are situatedalong the semi-minor axis (as depicted in Fig. 2d(ii))attract each other. There is, however, an important dif-ference between the two directions. In contrast to theprevious case, decreasing the asphericity of the ellipsoidmakes the attraction force between colloids weaker. Sincethe number of membrane beads adhered to each colloid

4

Adhesion Energy

Δφ

ΔΕ

/k T bb

ΔΕ

/k T b

Δφ

a)

c) d)

260

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-844

-841

-838

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0 π/4 π/83

0 π/4 π/83π/8 π/2

π/8 π/2

Δφ

Total Energy

Δφ

ΔΕ

/k T b

b)

1102

1107

1112

1117

1122

0 π/4 π/83π/8 π/2

Bending Energy

FIG. 3. Bending is the dominant term in the attraction of colloids. As shown in (b) and (d), both the adhesion energy andthe curvature energy are decreasing when the pair of colloids gets aligned with semi-minor axis of the ellipsoid. The latter hasa larger contribution in the total energy.

remains the same, this behavior cannot be explained bythe adhesion energy of the membrane.

To illuminate the reason that colloids select thedirection along the minor axis to attract each other,we investigate the energy of a pair of colloids along adifferent coordinate. As shown in Fig. 2d(iii), we rotatea pair of colloids, that are constrained at a fixed distanceto their center, along the angle spanning the spacebetween the semi-major and -minor axes. As Fig. 2cdepicts, the most energetically favorable configurationis when the colloids are aligned with the directionperpendicular to the major axis. This itself introducesa mechanism by which, without involving any otherfactors, two colloids find the mid-plane perpendicular tothe symmetry axes of the ellipsoid, as it minimizes thetotal energy of the membrane. In contrast, in the case ofhaving a perfect spherical membrane, it is not possibleto predict the localization of colloidal aggregates as itwill be randomly chosen. Increasing the major axisof the membrane (making e larger), drives the colloidreorientation stronger. One should be careful about thevalues for the bending moduli and adhesion coefficientsduring the simulations, as it can cause an effect wherecolloids are arrested and prevented from diffusing on thesurface of the membrane [23]. In addition, a very highvalue of KEll, in addition to influencing the adhesionenergy between the colloids and the membrane, wouldalso pull two tubes out of the vesicle.Although the above results quantitatively show differentbehavior in two directions, the dominant contribution inthe total energy of the membrane causing this effect isnot yet clarified. In order to approximately determine it,we proceed as follows: we pick a vesicle with e = 1.351and constrain the position of the colloids with a strongpotential. Here, in contrast to earlier, we do not use

the sampling method. Instead we let the system explorepossible configurations of the membrane after reachingequilibrium, and then take the average of the energiesfor all those configurations. As depicted in Fig. 3, boththe adhesion energy and the curvature of the membranedecrease when the angle between the line connecting twocolloids and the semi-major axis of the ellipsoid (Fig.3d) approaches π/2. The bending energy, as quantifiedin Fig. 3b, has a larger contribution to the total energythan the adhesion energy (Fig. 3c).Putting all the results together, we expect that when wehave more than two colloids they will initially attracteach other to form linear aggregates (to minimize theadhesion energy), and afterwards these aggregationschange their orientation to align with the minor axesof the ellipsoid. This is indeed what we observe in oursimulations. Fig. 4 depicts the equilibrium shape ofthe membrane for different numbers of colloids. In allthe test cases colloids tend to form a ring-like structurein the mid-plane of the ellipsoid. With a sufficientlylarge number of colloids (Figs. 4c and 4d), they form afull ring in this plane (see also the supplemental movieSM1). It is important to mention that these patterns arequite stable during the whole simulation. In contrast,in spherical vesicles there is no preferred direction forthe aggregation of particles. Although colloids attracteach other on a spherical membrane (Fig. 1), there isno preference for the direction of the attraction. Thismeans that even in case of forming a perfect ring on avesicle, particles self-assemble in an arbitrary directionon the membrane.As the final experiment, we look at the movement of

particles on a vesicle having negatively curved regions.To do so, we first overstretch the springs (by whichwe give quasi-ellipsoidal shape to vesicles) and form

5

a) b)

c) d)

FIG. 4. Colloids attract each other on ellipsoids, and in order to minimize the curvature energy, they form an arc (a & b) anda ring (c & d) at the mid-plane of the ellipsoid.

negatively curved regions in a big vesicle (Dv = 40σ).Having inserted a dimer in the system, we then look atthe migration of the dimer. As shown in Fig. 5 in thiscase the dimer does not stay at the mid-plane of thevesicle. It instead spends much of its time during MCsimulation at the regions that are negatively curved.Since the springs are overstretched, in the regions closeto poles there is no excess area for the dimer to adhereto and therefore the dimer cannot explore that area (seealso supplemental movies SM2-4). This type of dimermigration toward the areas having higher deviatoriccurvature is also experimentally observed in Ref. [31].

The type of pattern formation we observe in our simu-lations is reminiscent of recruiting proteins by the mem-brane during different biological processes. It has beenshown that, for example, dynamin proteins form a ringlike structure during exocytosis to facilitate membranescission [32] and that FtsZ proteins self-assemble intorings during the last step of bacterial cell division, namelycytokinesis [33]. Because most of the proteins in biologi-cal cells are either anchored to or embedded in the mem-brane, their interaction is a response to the deformationof the membrane they themselves impose. As in our sim-ulation the varying curvature is a determining factor thatdrives the pattern formation, we can relate our resultsto those membrane trafficking machinery functions. Al-though in this study we adjusted the included harmonicpotential strength KEll such that it would not affect theinteraction of the colloids with the membrane, it has beenproven that during the cell division we have the same sit-uation. Cytoplasmic dynein, as a multi-subunit molecu-lar motor, generates the force that is exploited by the cellto direct the orientation of the division axis by mitoticspindles [34]. Our results show that curvature inhomo-geneity and anisotropy can at least facilitate the processof protein self-assembly in the mid-plane of the cell.Although we have only investigated the interaction be-

tween identical isotropic inclusions, our results can ex-plain the behavior of a system containing anisotropicallyshaped inclusions as well. Based on the local deforma-tion of an ellipsoid, we expect that anisotropic inclusionsadhered to a spherical membrane attract each other inthe direction of negative curvature (with respect to thecurvature of the membrane). This situation correspondsto having an isotropic inclusion embedded in a membranewith an anisotropic shape, which is the case we have stud-ied here.

CONCLUSION

We studied the role of curvature heterogeneity andanisotropy on the interaction between colloids adheredto a membrane. First, we showed that the strength ofthe interaction between two colloids on the surface of aspherical vesicle is altered by changing the size of thevesicle. Next, we focused on such interactions on a mem-brane with an ellipsoidal shape. We revealed that theinteraction on such an inhomogeneously shaped mem-brane depends on direction. For example, decreasing theasphericity of an ellipsoidal membrane makes the attrac-tion between the colloids stronger along the semi-majoraxis and weaker in the semi-minor direction. Similarly,it has been previously shown in simulations that, on anelastic cylindrical membrane, colloids assemble perpen-dicularly to its major axis in the regime dominated bythe bending energy [25, 29]. In case of fluid membranes,through an analytical framework, it has also been shownthat inclusions “embedded” in a tubular membrane canattract each other in a transversal direction[20]. Simu-lating a vesicle containing many colloids, we showed howthey form a ringlike structure around the mid-plane ofthe ellipsoid. While the cluster of colloids freely exploresall the surface of a spherical membrane, less curved areaenergetically is more favorable for colloids on an ellip-soid. Our results suggest that forming regions of differ-

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0.08

0.16

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d c

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50

D

FIG. 5. Histogram of the geometric center of a colloid dimer on a vesicle (of size Dv = 40σ) with negatively curved regions(inset). The dimer spends most of the time in the negatively curved part of the vesicle.

ent curvatures on membrane vesicles can control patternformation of inclusions, and this can be important fromboth nanotechnological application and biological pointsof view.

ACKNOWLEDGEMENTS

This work was supported by the Netherlands Organ-isation for Scientific Research (NWO/OCW), as part ofthe Frontiers of Nanoscience program.

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