Nanoscale

PAPER

Cite this: Nanoscale, 2019, 11, 19751

Received 26th April 2019,Accepted 24th July 2019

DOI: 10.1039/c9nr03554k

rsc.li/nanoscale

Curvature-mediated cooperative wrapping ofmultiple nanoparticles at the same and oppositemembrane sides†

Zengshuai Yan,‡a Zeming Wu, ‡b Shixin Li,a Xianren Zhang, c Xin Yi *b andTongtao Yue *a

Cell membrane interactions with nanoparticles (NPs) are essential to cellular functioning and mostly

accompanied by membrane curvature generation and sensing. Multiple NPs inducing curvature from one

side of a membrane are believed to be wrapped cooperatively by the membrane through curvature-

mediated interactions. However, little is known about another biologically ubiquitous and important case,

i.e., NPs binding to opposite membrane sides induce a curved bend of different directions. Combining

coarse-grained molecular dynamics and theoretical analysis, here we systematically investigate the coop-

erative effect in the wrapping of multiple adhesive NPs at the same and opposite membrane sides and

demonstrate the importance of the magnitude and direction of the membrane bend in regulating

curvature-mediated NP interactions. Effects of the NP size, size difference, initial distance, number, and

strength of adhesion with the membrane on the wrapping cooperativity and wrapping states are analyzed.

For NPs binding to the same membrane side, rich membrane wrapping and NP aggregation states are

observed, and the curvature-mediated interactions could be either attractive or repulsive, depending on

the initial NP distance and the competition between the membrane bending, NP binding and membrane

protrusion. In sharp contrast, the interaction between two NPs binding to opposite membrane sides is

always attractive and the cooperative wrapping of NPs is promoted, as the curved membrane regions

induced by the NPs are shared in a manner that the NP–membrane contact is increased and the energy

cost of membrane bending is reduced. Owing to the ubiquity and heterogeneity of membrane shaping

proteins in biology, our results enrich the cutting-edge knowledge on the curvature-mediated interaction

of NPs for better and profound understanding on high-order cooperative assemblies of NPs or proteins in

numerous biological processes.

1 Introduction

The cell membrane not only serves as a physical barrier segre-gating the cell interior from the extracellular space, but also

plays a central role in many cellular functioning events invol-ving passive or active membrane remodeling such as endocyto-sis and exocytosis,1 membrane trafficking and signaling,2 andcell division and migration.3,4 Generally, membrane remodel-ing is barely an independent event, and involves generatingand sensing the membrane curvature.5,6 Mechanisms such asscaffolding, protein insertion and actin polymerization can beused by cells to induce membrane curvature.3,6,7 Oncedeformed, lipid molecules and proteins of different shapes aresorted or segregated in specific regions to stabilize the mem-brane curvature.6,8,9 Depending on the deformation extent oftwo curved membrane regions, either a repulsive or attractiveforce between objects could be induced by the curvature-mediated interaction.10–13 For the wrapping of multiple nano-particles (NPs), the curvature-mediated interaction becomesmore complicated and is involved in a wide range of cellularactivities including membrane tubulation and endocytosis ofmultiple NPs.14–22

†Electronic supplementary information (ESI) available: Details of the dissipativeparticle dynamics (DPD) simulation method, the N-varied DPD method, and thetheoretical analysis method; additional simulation results of the membranewrapping on NPs at the same and opposite sides; and additional data on thetheoretical modeling of membrane wrapping on two NPs at the same and oppo-site sides. See DOI: 10.1039/c9nr03554k‡Equal contribution.

aState Key Laboratory of Heavy Oil Processing, Center for Bioengineering and

Biotechnology, College of Chemical Engineering, China University of Petroleum

(East China), Qingdao 266580, China. E-mail: yuett@upc.edu.cnbDepartment of Mechanics and Engineering Science, Beijing Innovation Center for

Engineering Science and Advanced Technology, College of Engineering,

Peking University, Beijing 100871, China. E-mail: xyi@pku.edu.cncState Key Laboratory of Organic-Inorganic Composites, Beijing University of

Chemical Technology, Beijing 100029, China

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For NPs binding to the same membrane side, theequilibrium wrapping state is determined by the competitionbetween the membrane bending and NP adhesion.23–26 Theaggregation of multiple NPs reduces the membrane bendingdeformation as the NP–membrane adhesion is weakened.14–19

This is consistent with our previous simulations on the co-operative endocytosis of multiple NPs, which demonstratedthat relatively small NPs cluster into a closely packed aggregatewrapped by the membrane as a whole, whereas larger NPs areapt to separate and be wrapped independently.16,27

Although aggregation of NPs on the same side of a lipidmembrane has been observed in experiments,21,22,28,29 currentexperimental approaches have been proven to be inconclusivein quantifying curvature-mediated interactions underlying thecooperative effect. Theoretical modeling and simulations haveserved as a powerful complementary tool to experiments inthis regard and have been used quite effectively to reveal themechanism of NP aggregation on the membrane or the co-operative wrapping.11–21,27,30–33 Notably, the membrane defor-mation of different bend directions can be found frequently inbiological systems. In particular, NPs binding to the outer orinner sides of a cell membrane can initiate the processes ofendocytosis or exocytosis, respectively.15,17,25,34–39 Proteinscontaining BAR domains being intrinsically curved exhibit theability to induce membrane curvature (concave or convex),40,41

and the direction of membrane bend depends on the intrinsiccurvature of the domain. For example, classical BAR domainsinduce convex membrane deformation, while the I-BARdomain can induce concave membrane bending. The coexis-tence of these two types of BAR domains can thus generatemembrane curvatures of different bend directions, thus indu-cing the cooperativity of proteins.42 In addition, many BAR-domain proteins have links by way of signaling proteins toactin networks, the polymerization of which further inducescurvature nearby.43,44 Recently, the dynamic coupling betweenadjacent curvatures of different orientations was found toinduce membrane wave propagation involved in corticalprotein dynamics.45 Owing to the ubiquity and heterogeneityof membrane remodeling in biology, we investigated how thecurvature-mediated interaction is affected by the bend direc-tion of the deformed membrane in addition to the magnitudeof the bend.

Here we combine coarse-grained membrane simulationswith theoretical analysis to investigate the cooperative effect inNP binding at the same and opposite sides of a lipid mem-brane. Effects of the NP size, size difference, initial distance,number, and strength of adhesion with the membrane areanalyzed. In the case of two NPs binding to the same mem-brane side, the curvature-mediated interactions between NPscould be attractive or repulsive, depending on the initial NPdistance and the competition between membrane bending, NPbinding and membrane protrusion; while the interactionbetween and the net force on two NPs adhering to the oppositemembrane sides are always attractive as demonstrated. Ourstudies demonstrate the importance of curvature magnitudeand associated direction in membrane-mediated NP inter-

actions, and shed light on cooperative assemblies of NPs orproteins in many cellular processes.

2 Methods2.1 Coarse-grained models and the simulation method

The simulation system was prepared by positioning multipleNPs at the same and opposite sides of a lipid bilayer, as illus-trated in Fig. 1A. Each lipid was constructed by connectingthree hydrophilic beads (H) with two tails, each containing fivehydrophobic beads (T) (Fig. 1B). Such a coarse-grained model(H3T5), which represents dimyristoyl-phosphatidylcholine(DMPC), could form stable lipid bilayers with representativephase behaviors and has been widely used to investigate mem-brane interactions with NPs.16,27,46,47 Fifty percent of lipidmolecules were set to act as receptors (R) (Fig. 1C) with thefirst head bead as an active site capable of binding to ligands(L) coated on the NP surfaces.48–50 In this way, the receptordiffusion would not be a factor that limits the rate of mem-brane wrapping in our simulations.23,48,49 Each NP was con-structed by arranging hydrophilic beads (P) into a sphere of agiven diameter (Fig. 1D), and a half of the surface beads wereset as ligands, unless otherwise specified. During the simu-

Fig. 1 Schematic illustration of the simulation system setup (A), coarse-grained lipid and NP models in our simulations (B–D), and the theo-retical modeling (E and F). (A) Membrane interacting with multiple NPsat the same and opposite membrane sides. (B) Coarse-grained lipidmodel with three hydrophilic beads colored in green connected withtwo hydrophobic tails in yellow. (C) Coarse-grained receptor model withthe purple bead as the active site binding to ligands on the NP surfaces.(D) Spherical NP coated with ligand beads in white. (E and F) Symmetric/antisymmetric configuration of two identical rigid cylindrical NPs of dia-meter D adhering to the same/opposite sides of a membrane in theadopted two-dimensional Cartesian coordinate rz. Here ψ1 and ψ2 havepositive values.

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lations, each NP was restrained to move as a rigid body, andother components, including lipids and water molecules (W),were prevented from penetrating the NPs.

Simulations presented in this work are based on dissipativeparticle dynamics (DPD), which is a coarse-grained simulationtechnique taking into account hydrodynamic interactions.51–53

It has been extensively adopted as an effective and efficientcomputational method to study the mesoscale behaviors oflipid membranes and the NP–membrane interactions whileretaining the essential physical features of the modeledsystem.16,26,54–62 Here a specific variant of the DPD method,named the N-varied DPD simulation method,63,64 was appliedto simulate the membrane interaction with NPs. Briefly, usingthis method the targeted membrane tension can be main-tained constant by adjusting the lipid density ρLNPA in themembrane boundary region serving as a lipid reservoir. Unlessexplicitly stated otherwise, the lipid density ρLNPA was taken as1.7, which corresponds to a slightly negative membranetension as calculated in previous simulations.54 Such a smallmembrane tension is adopted to promote the membrane wrap-ping of NPs and save computation time. Details of the simu-lation method are described in the ESI.†

2.2 Theoretical modeling

For essential understanding on the interaction of NPs bindingto the same or opposite sides of the membrane, we performedtwo-dimensional theoretical analysis based on the Canham–

Helfrich membrane theory. We considered two cylindrical NPsof diameter D adhering on the same or opposite sides of amembrane (Fig. 1E and F). Here we present the case of NPsadhering on opposite membrane sides, and the detailedtheoretical results on the membrane wrapping of two cylindri-cal NPs at the same membrane side could be found in theESI.† As an antisymmetric configuration with respect to thepoint O in the adopted rz-coordinate is assumed in the case ofNPs at opposite membrane sides (Fig. 1F), our analysis isfocused on the right part of the system. The right part of themembrane can be divided into three parts: the inner freemembrane (blue curve, region 1), the adhesion region (redcircular arc), and the outer free membrane (black curve,region 2). Hereinafter subscripts 1 and 2 are used to identifyquantities associated with the inner and outer free membraneregions, respectively. The total free energy of the system perunit length in the out-of-plane direction is

E ¼ 2� κ

2

ðψ̇1

2dsþðψ̇2

2ds� �

þ σΔl þ fπκa� γ � 2πfa

� �;

where κ and σ denote the bending rigidity and tension of themembrane, respectively; ψ is the tangent angle of the mem-brane profile and the signed curvature is ψ̇ ≡ dψ/ds with s asthe arclength; the excess length Δl ¼ Ð ð1� cos ψÞds is conju-gated to the membrane tension σ; γ is the adhesion energy andf = (α + β)/(2π)∈ [0,1] is the wrapping degree defined as thelength ratio between the contact region and the circumferenceof the NP. Here α(β) represents the contact angle between thecylinder and the inner (outer) free membrane. The prefactor 2

in the above equation originates from equal energy contri-butions from the right and left parts of the system.

The minimum energy state of the vesicle at each givenwrapping degree f and NP distance d can be expressed as E =E(ψ1,ψ1,α), and is determined using the interior point optimi-zation technique65,66 with appropriate boundary conditionsas detailed in the ESI.† Then the local membrane force Fmem

per unit length in the out-of-plane direction can be deter-mined as67,68

Fmem ¼ Fitti þ Finni ¼ ðFit cos ψ i � Fin sin ψ iÞerþ ðFit sin ψ i þ Fin cos ψ iÞez;

where er and ez are mutually orthogonal unit basis vectors inthe r- and z-directions, respectively; ti = cos ψier + sin ψiez andni = −sin ψier + cos ψiez are the unit tangential and normalvectors of the free membrane profile, respectively, and thelocal force components Fi

t and Fin in the two-dimensional case

can be expressed as Fit = σ − κψ̇i

2/2 and Fin = −κψ̈i with ψ̈ ≡ d2ψ/ds2.

In the outer free membrane, numerical results indicate thatthe membrane forces in the r- and z-directions are F2

r =Fmem·er = σ and F2

z = Fmem·ez = 0, respectively, which can bedetermined more intuitively by drawing a free-body diagram ofa segment of the outer free membrane including the remotepoint of asymptotic membrane flatness and analyzing itsmechanical equilibrium. As Fr and Fz in the outer free mem-brane are known, in the following discussion we only focus onthe membrane forces F1

r = Fmem·er and F1z = Fmem·ez in the

inner free membrane which can only be determined numeri-cally. In our analysis, we are more interested in the differenceof the membrane force in the r-direction which regulates theNP aggregation. Therefore, we introduce ΔFr = F1

r − F2r = F1

r − σ.A positive ΔFr means that the two NPs tend to aggregate. Asshown theoretically in subsection 3.4, the interaction betweentwo identical cylindrical NPs binding to opposite membranesides is always attractive, consistent with previous two-dimen-sional theoretical studies.30–32

The two-dimensional theoretical modeling on the wrappingof two identical cylindrical NPs binding to the samemembrane side indicates that the NP interaction is alwaysrepulsive30–32 (see the ESI† for a detailed analysis).

3 Results and discussion3.1 Cooperative membrane wrapping of NPs at the same side

We started our simulations by positioning two NPs in closeproximity above the top surface of a modeled lipid membraneof lateral dimensions 52 nm × 52 nm. Sixty independent simu-lations were performed to investigate the wrapping cooperativ-ity and wrapping states, which depend on the NP size, NP dis-tance and receptor–ligand interaction strength (Fig. 2A and B).From these simulations, four different membrane wrappingand NP aggregation states were identified (Fig. 2C–F). Notethat the configuration in the upright corner of Fig. 2F is anintermediate state and the bottom subfigure in Fig. 2F rep-resents one of the four typical final configurations.

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For two NPs with a distance longer than the critical value,they were wrapped by the membrane independently (Fig. 2C).As the NP distance decreases, the deformed membraneregions induced by individual NPs overlapped to yield curva-ture-mediated interactions, determining whether two NPsaggregate to be wrapped by the membrane cooperatively orseparate to be wrapped independently. To reduce the storedmembrane bending energy, two NPs tend to aggregate and bewrapped by the membrane as a whole at the cost of decreasingcontact between receptors and ligands. In particular, the coop-erative wrapping can be regarded as the membrane wrappingtwo NPs together with the free energy smaller than twice theenergy of wrapping around one NP. Therefore, whether NPsaggregate or separate during the membrane wrapping essen-tially depends on the energetic competition between mem-brane bending and NP binding.16 Despite failure in calculatingthese energy items in simulations, we estimated the equili-brium state via monitoring changes in the NP distance andcorresponding wrapping configuration. In fact, the magnitudeof such membrane-mediated interactions between NPs adher-ing on the same membrane side has been calculated usingboth numerical and molecular dynamics simulations withthe consideration of the highly nonlinear nature of the

problem.12,13 Both attractive and repulsive forces wereobtained depending on the magnitude of the imprinted mem-brane curvature that can be modulated by the NP–membraneadhesion strength.12 Similar features of the variation betweenattractive and repulsive interactions have also been predictedin the theoretical study on the interaction of conical mem-brane inclusions as the membrane tension changes,10 inwhich a new length scale

ffiffiffiffiffiffiffiffiκ=σ

pin addition to the NP size is

introduced. In general, the membrane deformation on alength scale smaller than

ffiffiffiffiffiffiffiffiκ=σ

ppredominantly costs the

bending energy, while the membrane deformation on a largerscale is dominated by the membrane tension.69 As shown inFig. 2D, we observed aggregation of two NPs with D = 6.5 nmat the NP–membrane adhesion strength ε = 3.5 kBT and aninitial distance of 13 nm, indicating an attractive membrane-mediated interaction. As the adhesion strength ε becamelarger than 5.0 kBT, the two NPs were found to be wrapped bythe membrane independently (Fig. 2C). In this case, the strongreceptor–ligand interaction exceeds the energy cost of mem-brane bending, and NPs prefer to separate to maximize thecontact with the membrane during the dynamic wrappingprocess (Fig. S1†). Similar trends were observed for smallerNPs (Fig. 2B), or at a lower density of ligands (Fig. S2†).

Fig. 2 The equilibrium membrane wrapping and NP aggregation states as functions of the NP size, NP distance and receptor–ligand bindingstrength. (A, B) Phase diagrams of wrapping states on two NPs of diameter D = 6.5 nm (A) and D = 3.5 nm (B) with respect to the NP distance andreceptor–ligand binding strength. Symbols in A and B represent different wrapping configurations as selectively demonstrated in C–F. (C)Representative snapshot showing the independent membrane wrapping of two NPs of D = 6.5 nm at a relatively high receptor–ligand interactionstrength ε = 5.0 kBT and a relatively long NP distance. (D) Aggregation of NPs cooperatively wrapped by the membrane at a relatively smallε = 3.5 kBT. (E) Slight adhesion of two smaller NPs of D = 3.5 nm on the membrane at ε = 3.5 kBT. (F) Aggregation of two NPs separated by a curved mem-brane layer in between at ε = 5.0 kBT and a relatively short NP distance. An intermediate rather than the final wrapping state is displayed in the upright cornerof (F) for illustrating the generation of a membrane protrusion between two NPs. Each snapshot is displayed from both top and cross-sectional views.

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In particular, at ε = 3.5 kBT, two small NPs of D = 3.5 nm justadhered to the membrane barely inducing curvature due tothe low adhesion strength and the higher bending energy costfor wrapping smaller NPs (Fig. 2E). Thus, the membrane-mediated interaction can be ignored and two NPs freely movedon the membrane (Fig. S3†). As the NP size decreases, wefound an integral shift of the cooperative wrapping phasetoward conditions of higher interaction strength and smallerNP distance.

For two NPs with a relatively short NP distance and at ahigh receptor–ligand binding strength, a novel intermediatewrapping state was observed, in which a membrane protrusionwas formed and filled the gap between two NPs by formingtight contact with the NP surfaces (Fig. 2F). The membraneprotrusion in between and the membrane below the NPstogether formed a membrane junction, which was notobserved in the other three wrapping states (Fig. 2C–E). Toreveal the pathway and mechanism underlying the membranejunction formation, we performed additional simulations atε = 5.0 kBT and d = 9.4 nm (Fig. S4†). In the early stage, NPswere partially wrapped by the membrane simultaneously witha gap in between. Driven by the strong adhesion between themembrane and each NP, the membrane between NPs wasextracted to form a large area of contact with NP surfaces, anda protruding edge occurred at t = 3.5 μs, followed by a rapidincrease of the wrapping degree. The NP distance slightlyincreased to enable insertion of the protruding layer betweenNPs without causing a large compression energy due to themembrane thickness variation (see the inset of Fig. S4†). Asthe protrusion developed, one NP was fully wrapped first withthe wrapping of the other NP lagging behind, and eventuallyboth NPs were fully wrapped.

Due to the limited spatial and temporal resolution incurrent microscopy techniques, it remains challenging toexperimentally monitor or capture images of the membraneprotrusion formed in between two wrapped NPs in cell uptake.However, similar behaviors of the membrane protrusionaround NPs have been observed in other simulation studies onthe membrane wrapping of NPs driven by the strong adhesiveinteraction,18,70,71 which could arise from specific ligand–receptor bonding and non-specific van der Waals, steric, andelectrostatic forces at the NP–membrane interface. Theseresults together with our results demonstrated in Fig. 2F indi-cate that new wrapping and interaction modes involving mem-brane fusion could occur at strong adhesive NP–membraneinteractions.

3.2 Cooperative wrapping of NPs at opposite sides of amembrane

We next probe the wrapping of NPs from opposite sides of amembrane. Here we consider two NPs of diameter D = 6.5 nmrespectively above and below a membrane. According to thewrapping phase diagram (Fig. 3), two NPs slightly adhered andfreely moved on the membrane surfaces under rather lowreceptor–ligand binding strengths (e.g. ε < 3.0 kBT ), suggestingthat the NP induced membrane deformation can be ignored in

this case and there barely existed curvature-mediated inter-action driving the aggregation or separation of NPs. As ε

increased, two NPs were found to be wrapped by the mem-brane cooperatively or independently, depending on the NPdistance (Fig. 3). Once the NP distance exceeded the criticalvalue, which depends on ε, two NPs were wrapped by the mem-brane independently. Otherwise, they aggregated and werewrapped by the membrane cooperatively, albeit in a distinctpathway from that observed in two NPs binding to thesame membrane side (Fig. 2). At an extremely large ε (e.g.ε > 7.0 kBT ), NPs were rapidly wrapped by the membrane withthe formation of membrane protrusions,71 which restrainedNP aggregation as discussed in the following part.

To understand the effect of the NP distance on the pathwayand equilibrium state of NP wrapping from opposite mem-brane sides, we performed simulations at ε = 5.0 kBT andincreased the initial NP distance from 0 nm to 27.5 nm. A criti-cal value of the NP distance around 24 nm was estimatedbased on the results in Fig. 4, exceeding which two NPs werewrapped by the membrane independently. At the initial NPdistance d = 0 nm, NPs were firstly repelled to gain the mem-brane wrapping (Fig. 4A), and they became separated with thedistance rapidly increasing from 0 nm to 10.5 nm (Fig. 4F). Asthe simulation proceeded, the membrane wrapping of bothNPs drove two NPs to move around each other, as reflected bygradual decreases of both the NP distance (Fig. 4F) and theangle between the line connecting two NP centers and thehorizontal plane (Fig. 4G). When the initial NP distance wasclose to the sum of the membrane thickness and the NP dia-meter, both NPs can be wrapped by the membrane morerapidly (Fig. 4B and E), as the time for NP adjustment wassaved. In the case of a large initial NP distance of 20.7 nm, twoNPs were wrapped by the membrane independently with a flatmembrane region in between in the early wrapping stage

Fig. 3 The phase diagram summarizing the equilibrium states of themembrane wrapping of two NPs with D = 6.5 nm adhering to oppositesides of the membrane at different initial NP distances and adhesionstrengths. Five representative associated wrapping states listed in theright panel were identified from over sixty independent simulations. Forclarity, NPs initially locating above and below the membrane are dis-played in blue and red, respectively.

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(Fig. 4C, t = 2.4 μs). Then two NPs kept moving, respectively,upward and downward with an inclined and gradually curvedcentral membrane region. After completion of the cooperativemembrane wrapping, the NPs further reoriented and the lineconnecting two NPs rotated to a position parallel to the hori-zontal plane along with the membrane fusion between thewrapped regions and the outer free regions (Fig. 4G). If theinitial NP distance exceeds the critical value around 24 nm,two NPs were eventually wrapped by the membrane inde-pendently (Fig. 4D).

The equilibrium wrapping state for two NPs at oppositemembrane sides is also affected by the NP–membraneadhesion strength as depicted in Fig. S5.† It manifested thatwhether two NPs binding to opposite membrane sides can bewrapped cooperatively is determined by the degree of overlapbetween curved membrane regions wrapping on individualNPs. At ε = 3.0 kBT, the curved membrane regions were toosmall and hardly overlapped to mediate the interactionbetween two NPs (Fig. S5A†). As the initial NP distancedecreased below 15 nm, the two NPs were found to be partiallywrapped by the membrane cooperatively (Fig. S6†). Underhigher interaction strengths (e.g., ε = 4.0 kBT and 5.0 kBT ), twoNPs were found to be wrapped by the membrane to higherextents, thus increasing the overlap between curved regions toease cooperative wrapping on two NPs (Fig. S5B and C†). Asthe interaction strength was further increased to ε = 8.0 kBT,however, two NPs were rapidly wrapped by the membrane

before the completion of NP aggregation (Fig. S5D–F†). Afterthe completion of independent membrane wrapping on indi-vidual NPs, they further adjusted their locations along themembrane normal direction to further reduce the membranebending energy. No decrease of the NP distance was observed(Fig. S5F†), as further transition from the independent wrap-ping to the cooperative wrapping state must overcome a finiteenergy barrier arising from that mimicking vesicle fusion.72

To investigate the effects of membrane tension on the wrap-ping of NPs at opposite membrane sides, we increased themembrane tension by decreasing the target lipid density from1.7 to 1.65 and 1.6 (Fig. S7 and S8†). The initial NP distancewas taken as d = 20.7 nm and the NP diameter D = 6.5 nm. Atε = 5.0 kBT, the cooperative wrapping slowed down by increas-ing the membrane tension (Fig. S7†). This could be under-stood as follows. As the membrane tension increased, themembrane bending became more difficult and the degree ofcurvature overlap mediating the cooperative wrapping wasweakened. At a stronger interaction strength ε = 8.0 kBT(Fig. S8†), the effect of membrane tension on the wrappingcooperativity became different. In that case, NPs at a smallermembrane tension were wrapped so rapidly that they achievedthe full wrapping states individually before significant aggrega-tion. The cooperative wrapping was hindered. As the mem-brane tension increased, the wrapping slowed down andbecame difficult even at such high interaction strength (ε =8.0 kBT ), leaving more time for both NPs to approach and

Fig. 4 Effects of the initial NP distance on the membrane wrapping of two NPs from opposite sides of a membrane. (A–D) Time sequences oftypical snapshots at initial NP distances: 0 nm (A), 13.8 nm (B), 20.7 nm (C) and 27.5 nm (D). Corresponding time evolutions of the wrapping degree(E), NP distance (F), and angle between the line connecting NP centers and the horizontal plane (G). Here we take ε = 5.0 kBT.

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accomplish the cooperative wrapping. From Fig. S7 and S8,†we could see that the wrapping cooperativity is regulated bythe rate of NP wrapping and NP aggregation. A sufficientlystrong interaction strength together with a relatively smallmembrane tension hindered the cooperative wrapping byenabling fast individual and independent wrapping; other-wise, two NPs aggregated, interplayed and became wrapped bythe membrane cooperatively.

In the above analysis, two NPs have the same size. Here wecarry out further investigation to explore how a membranewraps two NPs of different sizes from opposite membranesides. Two NPs of different diameters D = 4 nm and 9 nm wereconsidered. The initial NP distance increased from 0 nm to36.8 nm with ε = 5.0 kBT. When these two NPs were initiallyarranged face-to-face (d = 0 nm), the larger NP was preferen-tially wrapped by the membrane due to the larger surface areaand the smaller curvature, whereas the smaller one kept adher-ing to the deformed membrane region during the finite simu-lation time (Fig. 5A, D, E and H). A large particle curvaturerestrained the membrane wrapping of the small NP (Fig. 5D).34

We expected that after a sufficiently long simulation time, thesmaller NP would diffuse to the curved region adjacent to thewrapped larger NP to increase the area of contact between theNPs and membrane and to reduce the energy cost of mem-brane bending energy. In the intermediate range of the initialNP distance, the smaller NP readily fell into the curved regioninduced by membrane wrapping around the larger NP. BothNPs were thus wrapped by the membrane cooperatively(Fig. 5B and D). The trajectories of two NPs showed that thelarger NP moved primarily along the vertical direction

(Fig. 5H), whereas the smaller one moved laterally around thelarger NP to promote the cooperative wrapping (Fig. 5B, Fand H). If two NPs were initially far away from each other, theywere wrapped independently as expected (Fig. 5C, D, G and H).

3.3 Equilibrium wrapping and arrangement of multiple NPsat both the same and opposite sides of a membrane

We extended our simulations by considering more NPsbinding to the opposite sides of the membrane. In the case offour adhesive NPs placed at four corners of a square with aside length 13 nm, these NPs at ε = 5.0 kBT were cooperativelywrapped by the membrane in a manner involving the mem-brane deformation and protrusion (Fig. 6A). For these NPsadhering to the same membrane side, they were wrapped bythe membrane via generation of a membrane protrusion inbetween, which adhered onto the remaining two NPs at theopposite side. Four NPs formed a tight interlocked configur-ation eventually. As the NP distance increased to 26 nm, allfour NPs were wrapped by the membrane independently(Fig. 6B). Next, we increased the wrapping complexity by posi-tioning six NPs with a short NP distance (6.5 nm) at two oppo-site membrane sides. Besides the interlocked wrapping con-figuration, six NPs were found to form an ordered arrangementof triangular lattices (Fig. 6C). Three NPs from the upper mem-brane side were cooperatively wrapped with the formation of acentral membrane protrusion. At a longer NP distance of13 nm, we found a synchronous contraction of all six NPs toform an inter-connected circular arrangement. Each two adja-cent NPs were wrapped cooperatively from opposite sides ofthe membrane, while no cooperation occurred for two NPs

Fig. 5 Membrane wrapping of two NPs with diameter D = 4 nm and 9 nm. Time sequences of typical snapshots at different initial NP distances:0 nm (A), 18.3 nm (B) and 36.8 nm (C). (D) Time evolutions of the membrane wrapping degree. (E–G) Trajectories of two NPs in the horizontal planeat different initial NP distances: 0 nm (E), 18.3 nm (F) and 36.8 nm (G). (H) Time evolutions of the NP positions along the vertical direction. The graylines in D and H represent the results of smaller NPs. The receptor–ligand interaction strength is ε = 5.0 kBT.

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binding to the same membrane side due to the longer distance(Fig. 6D).

Note that dynamic traps may exist to affect the equilibriumNP arrangement.27 We performed five independent simu-lations under each condition for the cases of six NPs at twoinitial NP distances (d = 6.5 nm and 13 nm) and three NP–membrane adhesion strengths (ε = 4.0 kBT, 5.0 kBT, and7.0 kBT ). All thirty final configurations are summarized inFig. S9.† It demonstrates that the final configurations could bedifferent even for the same parameter studies. Moreover, itshows that the equilibrium arrangements in short and longinitial distances are different. These observations indicate thatthere might exist dynamic traps associated with the membranewrapping and lateral motion of NPs inducing the disorderedarrangement of NPs during wrapping at both the same andopposite sides of a membrane. Nevertheless, the ordered NParrangement of triangular lattices is the most stable state formultiple NPs of a relatively short NP distance, while the inter-connected circular arrangement is more favorable for NPs of alonger initial distance. The dependence of the final configur-ation on the initial state has also been predicted in the theore-tical study on the interaction of conical membrane inclusions,which indicated that the inclusions of the opposite orientationrepel each other at small separations, but attract each other atlarger ones.10

Note that in our simulations we focused on the cases thatNPs came in contact with the membrane simultaneously,while in real biological environments the contact occurred

stochastically. Previous molecular dynamics simulations onthe cooperative wrapping of NPs from the same membraneside demonstrated that the order of NP–membrane contactcould affect the system configurations, depending on the NPsize, shape and numbers; however, the cooperative effect isshared in the cases of simultaneous and stochastic or sub-sequent contact.18 A thorough study on how the wrappingstates depend on the NP–membrane contact order and sidecertainly deserves further detailed investigation in the future.

3.4 Theoretical analysis on membrane wrapping of NPs atopposite sides

Simulations in previous sections have convincingly showedthat the wrapping and aggregation states of NPs from oppositesides of the membrane are distinct from that of NPs bindingto the same membrane side. While the curvature-mediatedinteraction between two NPs adhering to the same membraneside can be attractive or repulsive depending on the compe-tition between membrane bending and NP binding, the forcebetween two NPs from opposite membrane sides can be alwaysattractive. According to the state of membrane wrapping ontwo NPs from the opposite sides, the curved membraneregions induced by individual NPs are shared in the manner ofan antisymmetric configuration that the NP–membranecontact is increased and the energy cost of membrane bendingis reduced. Therefore, no energy barrier is observed during theaggregation of NPs from opposite membrane sides and thecooperative wrapping state is favorable in simulations.Moreover, this feature associated with the antisymmetric con-figuration is supposed to be independent of the dimensions ofthe system.

For further essential and rational understanding on theinteractions of NPs at opposite membrane sides, here weperform a two-dimensional theoretical analysis based on theCanham–Helfrich membrane theory (see subsection 2.2). Weconsider two identical cylindrical NPs of diameter D adheringto opposite sides of a membrane of bending rigidity κ andtension σ, and numerically determine the membrane elasticenergy and the membrane force as functions of the wrappingdegree f and the NP distance d. Different from previous theore-tical studies on a similar topic,30–32 our computational resultsnumerically span a full range from small to large membranedeformation while requiring the position of the right and leftouter membranes at the remote boundaries to be in the samehorizontal plane. At d > D, there is no contact between thesetwo NPs, and the energy increases slowly as d increases at thesame wrapping degree f (Fig. 7A). The smooth energy profilemeans that the wrapping process is continuous at d > D. Asd/D further decreases and falls into the range of [0,1], NPscontact and there exists an energy valley. We also analyze thelocal membrane force at d > D (Fig. 7B). It is indicated that thenet membrane force in the r-direction is positive and leads theNPs toward each other, consistent with the energy profile atd > D as analyzed above. The theoretically predicted profile ofthe curvature-mediated NP interactive force in Fig. 7B is con-sistent with our simulation results that the interaction

Fig. 6 Initial and final configurations of the membrane wrapping offour (A and B) and six NPs (C and D) adhering to opposite sides of themembrane. The nearest initial NP distances were 13 nm (A), 26 nm (B),6.5 nm (C), and 13 nm (D). The right panel is the cross-sectional viewsalong red lines. Arrows represent the locations of other wrapped NPs.Here we set ε = 5.0 kBT.

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between two NPs binding to opposite membrane sides isalways attractive and the cooperative wrapping of NPs is pro-moted (Fig. 4 and 5). Similar energy and local force profiles ata small membrane tension σ = 0.2κ/D2 can be found inFig. S10.† As σ increases, the maximum value of ΔFr increases.

Selected membrane configurations at σ = 6κ/D2 are shownin Fig. 8. At a given NP distance d > D, the magnitude of thevertical distance between two NPs decreases first and thenincreases as f increases. In the case of d/D∈ [0,1], the two sep-arated NPs come into contact with one another smoothly asthe wrapping proceeds (around f = 0.1 as shown in Fig. 8C), fol-

lowed by a discontinuous configurational transition in whichthe NP of an initial lower z-coordinate is inverted to a positionof a higher z-coordinate. This discontinuous configurationaltransition occurs around f = 0.38 and is reflected in the kinkedred line in Fig. S11.† Eventually two NPs reach a full wrappingstate around f = 0.87 in which the outer free membranes touchthe wrapped region. Similar configurations as predicted inFig. 8 are also demonstrated in our molecular dynamics simu-lations (Fig. 4).

In the continuum theoretical framework, the effectiveadhesion energy due to the specific receptor–ligand binding is

Fig. 7 (A) The system elastic deformation energy ED/κ (γ = 0) and (B) difference of local membrane force between the inner free and outer freemembrane in the r-direction. ΔFrD2/κ as a function of the wrapping degree f and particle distance d/D. Insets: Corresponding contour plots. Hereσ = 6κ/D2 is considered.

Fig. 8 Selected membrane configurations in the case of σ = 6κ/D2 at d/D = 1.5 (A), 1 (B) and 0.5 (C). Arrows at the NP centers represent the inter-action force ΔFr due to the membrane deformation with the force values listed on the right sides. A positive ΔFr indicates attractive NP interaction.

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captured by the adhesion energy γ which is proportional to theadhesion strength and density of the receptor–ligand bonds.For a certain adhesion energy γ, the most energetically favor-able configuration is determined by the minimum value of thetotal free energy E. For a small adhesion energy (gray lines inFig. 9), NPs are weakly wrapped and the energy curve containsa minimum energy point at d/D = 1. As γ increases, theminimum energy state occurs at a smaller d. Plots of thecorresponding wrapping degree f as a function of the NP dis-tance d/D can be found in Fig. S12.†

From a mechanical point of view, the membrane shapingby the adhesive NP is similar to the modification of the localmembrane shape by the segregation of lipids with specificshapes such as (inverted) cones or by cytoplasmic proteinmachineries such as wedge-shaped transmembrane proteins,protein oligomerization and the Bin-amphiphysin-Rvs (BAR)domain protein superfamily. As demonstrated by our simu-lations, the nature of the curvature-mediated interaction deter-mines whether curved proteins aggregate to be wrapped co-operatively by the membrane as a whole or they separate to bewrapped independently. In previous experiments, organizedassemblies of BAR proteins on the membrane were observed,which can be attributed to the curvature-mediated inter-action.40 There are also biological processes relevant to themembrane wrapping of NPs from opposite sides. For example,recent theoretical studies indicate that the aggregation ofmembrane inclusions into a cluster could induce membraneinvagination, and the interaction between the invaginations ofdifferent outward normal directions is attractive,73,74 consist-ent with our current simulations and theoretical results on theattractive interaction between NPs adhering at opposite mem-brane sides. As NPs undergoing cellular internalization andoutward membrane protrusions can be approximatelyregarded as adhering NPs on the opposite membrane sidesfrom a physical point of view, an inspiring and interestinghypothesis is that more NPs might be engulfed around thebase of membrane protrusions.

Regarding the membrane wrapping of two NPs at the samemembrane side, the current and previous two-dimensional

theoretical analysis indicated that the NP interaction is alwaysrepulsive (Fig. S13–S16†);30–32 while the three-dimensionaltheoretical work on the membrane-mediated interaction oftwo spherical NPs indicates that the NP interaction dependson the wrapping degree and the initial NP distance,11–13 andthe interaction could be repulsive or attractive, consistent withour DPD simulations. Recalling that only one nonzero princi-pal curvature exists in the two-dimensional model and twoprincipal curvatures in three-dimensional analysis, the inter-action difference between the wrapping of cylindrical andspherical NPs at the same side could be attributed to the localmembrane deformation or the membrane stress tensor,specifically the difference between squares of two principalcurvatures.12,67,68 A thorough and precise three-dimensionaltheoretical analysis on the membrane wrapping of multipleNPs is challenging and requires further investigation.

4 Conclusions

We have performed systematic coarse-grained simulationscombining theoretical analysis to investigate the cooperativeeffects in the wrapping of multiple NPs at the same and oppo-site sides of a modeled plasma membrane. For two NPsbinding to the same membrane side, they can be wrappedcooperatively or independently, depending on the NP size, NP–membrane adhesion strength and initial NP distance. Thecompetition between the membrane bending and NP adhesionis thought to determine whether the curvature-mediated inter-action is attractive or repulsive. Under certain conditions ofshort NP distance and strong receptor–ligand interaction, twoNPs could be wrapped by the membrane via the membraneprotrusions in between. That interaction mode is thought toincrease contact between NPs and the membrane at minimalenergy cost of the membrane deformation. In contrast, theinteractions between and the net forces on two NPs binding toopposite membrane sides are always attractive in definedranges of the NP distance. Simulations revealed that once thecurved membrane regions wrapping two NPs from opposite

Fig. 9 Total free energy E at different adhesion energies γ in the cases of σ = 0.2κ/D2 (A) and 6κ/D2 (B). Here σ̄ ≡ σD2/κ and γ̄ ≡ γD2/κ.

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sides overlap, two NPs readily shared the same curved mem-brane region to promote wrapping of both NPs and reduce theenergy cost of membrane bending. For two NPs of differentsizes, wrapping of the larger NP always prevails to promotewrapping of the smaller one. We further extended simulationsby considering more NPs adhering on both membrane sidesand found interlocked configurations manifesting the coopera-tive effects in the wrapping of multiple NPs from one and twomembrane sides. Our results enrich the knowledge of the cur-vature-mediated interaction between NPs and highlight theimportance of binding sides in regulating NP interactions,which may have important implications on the mechanism ofmembrane-mediated interactions between proteins or proteinoligomers as well as membrane shaping in endocytosis.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work is supported by the National Natural ScienceFoundation of China (No. 31871012 and 11872005), theNatural Science Foundation of Shandong Province (No.ZR2018MC004), and the Fundamental Research Funds for theCentral Universities (No. 19CX07002A).

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Supplementary Information for “Curvature-mediated cooperative wrapping of multiple nanoparticles at the same and opposite membrane sides”

Zengshuai Yan1,†, Zeming Wu2,†, Shixin Li1, Xianren Zhang3, Xin Yi2,*, Tongtao Yue1,*

1State Key Laboratory of Heavy Oil Processing, Center for Bioengineering and Biotechnology, College of Chemical Engineering, China University of Petroleum (East China), Qingdao 266580, China 2Department of Mechanics and Engineering Science, Beijing Innovation Center for Engineering Science and Advanced Technology, College of Engineering, Peking University, Beijing 100871, China 3State Key Laboratory of Organic-Inorganic Composites, Beijing University of Chemical Technology, Beijing 100029, China †Equal contribution *Address correspondence to Xin Yi, xyi@pku.edu.cn; Tongtao Yue, yuett@upc.edu.cn

Note 1: Dissipative particle dynamics simulation method

In our DPD simulations, the dynamics of elementary units obeys Newton’s law of

motion. Typically, beads i and j interact with one another via a pairwise additive force

consisting of the conservative force CijF , dissipative force D

ijF , and random force RijF .

The total force exerted on bead i can thus be expressed as C D R( )i ij ij ij

i jF F F F

≠

= + +∑ .

The conservative force CijF between beads i and j is of a soft repulsion acting

along the line connecting the bead centers and has the form C

cmax{1 / ,0}ij ij ij ijF a r r r= − ,

where aij is the maximum repulsive strength between beads i and j, rij=rj - ri (ri and rj

are their positions), ijr =rij /|rij|, and rc is the cut off radius and taken as rc =0.646 nm. In

the simulations, physical quantities are scaled with the cutoff radius rc, bead mass m,

and thermal energy kBT with kB as the Boltzmann constant. All simulations were

performed with the timestep of △t = 16 ps, with the periodic boundary conditions

adopted in all three directions.

According to previous studies,1,2 the values of interaction parameters aij between

beads of the same type in the pure membrane system were set to WWa =

H HHH R R 25a a= = and T TTT R R 15a a= = , and those between beads of different types

were TTW R W 80a a= = ,

T H H THT HR R T R R 50a a a a= = = = , and TTR 15a = . Detailed

values of the interaction parameters are summarized in Table S1.

The dissipative force is determined by the equation D 2

c(1 / ) ( )ij ij ij ij ijF r r r v r= −Γ − ⋅ ,

where Γ is the friction coefficient, vij = vi - vj (vi and vj are their velocities). This

expression conserves the momentum of each pair of interacting beads, and

consequently the total momentum of the system is conserved.

The random force between beads i and j is calculated by R 2

c(1 / )ij ij ij ijF r r r= −Σ − θ ,

where Σ represents the noise amplitude and ijθ is an uncorrelated random variable

with zero mean and unit variance.

For lipid molecules, the interaction between neighboring beads in the same

molecule is described by a harmonic spring force, which is given by

S S eq( )ij ijF K r r r= − ,

where KS = 128 is the spring constant and req = 0.7 is the equilibrium bond length.

In order to maintain the bending rigidity of lipids, the force constraining the

variation of the bond angle is calculated by

F Uϕ ϕ= −∇ and 0[1 cos( )]U Kϕ ϕ ϕ ϕ= − − ,

where 0 =ϕ π is the equilibrium bond angle, ϕ is the real bond angle, and K = 10.0

is the bond bending force constant.

In addition, to represent specific interactions between ligands coating on the NPs

and active points of receptors embedded in the membrane, a modified truncated LJ

potential was applied and defined by 12 6

LJ =4 ( / ) ( / ) 0.22ij ijU r r − + ε λ λ ε ,

where rij < rc, λ = 0.624rc, and ε represents the strength of the ligand-receptor

interaction. The largest repulsive force is set to be 10kBT/rc, aiming to ensure the propel

running of DPD simulations.3-5

Table S1 Interaction parameters (a) used in our simulations

H T RH RT P L W

H 25 50 25 50 25 25 25

T 50 15 50 15 80 80 80

RH 25 50 25 50 25 25 25

RT 50 15 50 15 80 80 80

P 25 80 25 80 25 25 25

L 25 80 25 80 25 25 25

W 25 80 25 80 25 25 25

Note 2: N-varied DPD method

In this work, a specific variant of the DPD method, named the N-varied DPD

simulation method, was applied to simulate the membrane interaction with NPs. In this

method, the targeted membrane tension can be maintained constant by adjusting the

lipid number per area (LNPA) in the membrane boundary region serving as a lipid

reservoir.6-8 By adding and removing lipids, the value of LNPA in the boundary region

is kept within a desired range ( min maxLNPA LNPA LNPAρ ρ ρ< < ). To keep the overall average

density of beads in the simulation box constant, water beads are added into or deleted

from the box correspondingly. Each addition or deletion move is performed every 1500

steps to leave enough time for the propagation of the membrane tension to the whole

membrane.

Note 3: Determination of the membrane configurations in theoretical model

In our theoretical modeling, the membrane configuration is fully characterized by the

tangent angle ψ with geometrical relations d / d cosr s ψ= and d / d sinz s ψ= , and

the minimum energy state of the vesicle at each given wrapping degree f and NP

distance d can be expressed as 1 2( , , )E E ψ ψ α= . Here we further introduce two new

variables /i i it s l= ( [0,1]it ∈ , i = 1, 2) with l1 and l2 as the total lengths of the inner

and outer free membranes, respectively, and reparametrize the unknown variables ψi(si)

as ( ) ( )( ) ( )i ii i j j it a N t=∑ψ (j = 0,1,…,ni) based on cubic B-spline approximation. Here

the control points ( )ija are the coefficients of the basic functions ( ) ( )i

j iN s .

We employ the interior point optimization technique to numerically determine the

minimum state of the membrane elastic energy at given f and d. The required boundary

conditions are as follows. In the case of two cylindrical NPs adhering on the same side

of the membrane, we have 0r z= = and 1 0ψ = at 1 0t = due to the symmetric

membrane configuration; while in the case of two cylindrical NPs adhering on the

opposite membrane sides, we have 0r z= = and zero membrane curvature ( 1 0=ψ )

at 1 0t = (point O) due to the antisymmetric membrane configuration. At the remote

boundary ( 2 1t = ), we have 0z = and 2 0ψ = . Other mandatory conditions are

continuities of the r-coordinate and tangent angle ψ at two contact edges. To

approximate the remote condition 2l →∞ , the total length l2 of the outer membrane

region is determined by a large prescribed length of the membrane projection of 20a.

Under these conditions, the system free energy E at given f and d is minimized with

respect to ( )ija , li, and α, and the corresponding membrane configuration could be

determined.

Note 4: Supplementary simulation results

Fig. S1 Time sequences of typical snapshots and evolutions of the NP distance for two identical NPs of diameter D = 6.5 nm at an initial distance of 13 nm but different receptor-ligand interaction strength ε.

Fig. S2 Equilibrium wrapping configurations as functions of the NP distance and receptor-ligand binding strength. (A-C) Typical snapshots from top and side views showing three representative membrane wrapping states. (D, E) Phase diagrams of wrapping states for two NPs of diameter D = 6.5 nm (D) and 3.5 nm (E) at different NP distances and receptor-ligand binding strengths. Symbols representing the wrapping states are illustrated in A-C. The density of ligands on the NP surface is decreased to 25% of that in Fig. 2 in the main text.

Fig. S3 Time evolution of the distance between two identical NPs weakly adhering on the membrane. The NP diameter is D = 3.5 nm and the receptor-ligand interaction parameter is ε = 3.5 kBT. Four typical snapshots at different time are presented in the right panel to demonstrate the NP movement on the membrane.

Fig. S4 Time evolutions of the wrapping degree for two NPs of D = 6.5 nm at ε = 5.0 kBT and two different initial NP distances (d = 9.4 nm and 17.2 nm). Insets show representative snapshots and the evolution of the NP distance in the time range between 2.5 μs and 4.0 μs.

Fig. S5 The effect of adhesion strength on the wrapping process of two NPs at opposite sides of the membrane. (A-D) Time sequences of typical snapshots of the membrane wrapping processes at ε = 3.0 kBT (A), 4.0 kBT (B), 5.0 kBT (C) and 8.0 kBT (D). Corresponding time evolutions of the average membrane wrapping degree (E) and NP distance (F). The initial NP distance was set to be 20.7 nm.

Fig. S6 The effect of the initial NP distance on the membrane wrapping of two NPs with diameter D = 6.5 nm at opposite membrane sides. (A-C) Time sequences of typical wrapping snapshots with initial NP distances as 0 nm, 13.8 nm, and 20.7 nm, respectively. (D) Time evolutions of the degree of membrane wrapping around each NP. NPs at shorter initial distances are wrapped by the membrane with a higher extent due to the cooperative effect. (E) Time evolutions of the NP distance. The receptor-ligand interaction strength was set as ε = 3.0 kBT.

Fig. S7 Effect of the membrane tension on the membrane wrapping of two NPs from opposite sides of the membrane at ε = 5.0 kBT. (A-C) Time sequences of typical snapshots at fixed lipid surface densities of 1.7 (A), 1.65 (B) and 1.60 (C). Time evolutions of the average membrane wrapping degree (D) and NP distance (E) at different lipid densities. The NP diameter and initial NP distance are D = 6.5 nm and d = 20.7 nm, respectively.

Fig. S8 Effect of membrane tension on the membrane wrapping of two NPs from opposite sides of the membrane at ε = 8.0 kBT. (A-C) Time sequences of typical snapshots at fixed lipid surface densities of 1.7 (A), 1.65 (B) and 1.60 (C). Time evolutions of the average membrane wrapping degree (D) and NP distance (E) at different lipid densities. The NP diameter and initial NP distance are D = 6.5 nm and d = 20.7 nm, respectively.

Fig. S9 Equilibrium NP arrangements at different initial NP distances and NP-membrane adhesion strengths. Under each condition five independent simulations were performed.

Note 5: Supplementary theoretical results

Membrane wrapping of two cylindrical NPs at the opposite membrane sides

Fig. S10 Energy and force profiles for the membrane wrapping of two cylindrical NPs at the opposite membrane sides. (A) The system elastic deformation energy ED/κ (at γ = 0) and (B) difference of local membrane force ΔFrD2/κ between the inner free and outer free membranes in the r-direction as functions of the wrapping degree f and NP distance d/D. Insets: corresponding contour plots. The membrane tension

20.2 / D=σ κ is considered here.

Fig. S11 The system elastic deformation energy ED/κ at zero adhesion energy as functions of the wrapping degree f at 26 / D=σ κ and different NP distance d/D in the case of the wrapping of two cylindrical NPs at the opposite membrane sides. Symbols on the curves represent the full wrapping states.

Fig. S12 Wrapping degree f at different adhesion energy γ in the cases of

20.2 / D=σ κ (A) and 26 / Dκ (B) for the wrapping of two cylindrical NPs at the opposite membrane sides. Here 2 /D≡σ σ κ and 2 /D≡γ γ κ .

Membrane wrapping of two cylindrical NPs at the same membrane side

For the wrapping of two identical cylindrical NPs of diameter D at the same

membrane side, the system energy and interaction force are basically the same as these

used for the wrapping of NPs at the opposite membrane sides (see subsection 2.2 in the

main text). At d > D, the elastic deformation energy decreases slowly as d increases at a

given f (Fig. S13A). The local membrane force rF∆ is negative and drives NPs away

from each other (Fig. S13B), indicating a repulsive NP interaction.

Fig. S13 (A) The system elastic deformation energy ED/κ (γ = 0) and (B) difference of local membrane force between the inner free and outer free membrane in the r-direction ∆FrD2/κ as functions of the wrapping degree f and NP distance d/D. Insets: corresponding contour plots. Here 26 / D=σ κ is considered.

Selected membrane configurations at 26 / D=σ κ are shown in Fig. S14. In the

case of / (1, 2)d D∈ , the wrapping of two separated NPs is smooth in the early

wrapping stage, followed by a discontinuous configurational transition with the

shallow wrapping state abruptly jumping to a deep wrapping state (as demonstrated in

Fig. S14A from f = 0.3 to f = 0.6). This discontinuous configurational transition occurs

around f = 0.5 and is reflected by the kinked red line in Fig. S15. Eventually two NPs

reach a full wrapping state around f = 0.9 in which the outer free membranes touch the

inner free membrane. The smooth blue short dash line in Fig. S15 means that the

wrapping process is continuous at d/D = 2.

Fig. S14 Selected membrane configurations at different f and d in the cases of

26 / D=σ κ . Arrows at the NP centers represent the interaction force rF∆ due to the membrane deformation with the force values listed on the right sides. A negative rF∆ indicates repulsive NP interaction.

Fig. S15 The system elastic deformation energy ED/κ at γ = 0 as functions of the wrapping degree f at 26 / D=σ κ and different NP distance d/D. Symbols on the curves represent the full wrapping states.

Fig. S16 Total free energy E (A) and wrapping degree f (B) at 6σ = and different γ .

At a given adhesion energy γ, the total free energy E decreases as the NP distance

d/D increases (Fig. S16A), indicating a repulsive NP interaction. The evolution of the

wrapping degree f as a function of d/D is shown in Fig. S16B.

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