The fusion of Membranes and Vesicles: Pathway and Energy ... · The Fusion of Membranes and...

The Fusion of Membranes and Vesicles:

Pathway and Energy Barriers from Dissipative

Particle Dynamics

Andrea Grafmuller1

Theory and Bio-Systems,Max Planck Institute for Colloids and Interfaces, Potsdam, Germany

Julian ShillcockMEMPHYS,

University of Southern Denmark, Odense, Denmark

Reinhard LipowskyTheory and Bio-Systems,

Max Planck Institute for Colloids and Interfaces, Potsdam, Germany

1Corresponding author. Address: Theory and Bio-Systems, Max Planck In-stitute for Colloids and Interfaces, Am Muhlenberg, Golm, 14476, Germany,Tel.: (0331)567-9620

Abstract

The fusion of lipid bilayers is studied with dissipative particle dynamics(DPD) simulations. First, to achieve control over membrane properties, theeffects of individual simulation parameters are studied and optimized. Then,a large number of fusion events for a vesicle and a planar bilayer is simu-lated using the optimized parameter set. In the observed fusion pathway,configurations of individual lipids play an important role. Fusion starts withindividual lipids assuming a splayed tail configuration with one tail insertedin each membrane. In order to determine the corresponding energy barrier,we measure the average work for interbilayer flips of a lipid tail, i.e., theaverage work to displace one lipid tail from one bilayer to the other. Thisenergy barrier is found to depend strongly on a certain DPD parameter, and,thus, can be adjusted in the simulations. Overall, three sub-processes havebeen identified in the fusion pathway. Their energy barriers are estimatedto lie in the range 8 − 15 kBT . The fusion probability is found to possess amaximum at intermediate tension values. As one decreases the tension, thefusion probability seems to vanish before the tensionless membrane state isattained. This would imply that the tension has to exceed a certain thresholdvalue in order to induce fusion.

Key words: membrane fusion; membrane tension; computer simulations;fusion probability; energy barrier; interbilayer flips

Vesicle Fusion from DPD 2

Introduction

Fusion of biological membranes is an essential process in many areas of cell bi-ology, ranging from vesicular trafficking and synaptic transmission to cell-cellfusion or viral fusion. Biological membranes are complex systems composedof many different lipids and proteins. For a better understanding of the fun-damental processes involved, lipid vesicles are often used as simplified modelsystems (1). Even in the absence of proteins, such model membranes can beinduced to fuse experimentally by a variety of methods.

For a fusion pore to form, drastic topological rearrangement of the twomembranes and a destruction of their bilayer structure is necessary at leastlocally. On the other hand, lipid bilayer membranes in water are very stablestructures that do not easily form holes. This makes the fusion process andits energetics an interesting problem, which has received much attention inrecent years.

The initial fusion pore is believed to be a neck-like connection with aninitial size of about 10 nm. The corresponding time scale has not beendirectly measured, but both patch clamp methods applied to synaptic mem-branes and ultrafast optical microscopy of giant vesicles suggest that thefusion pore can be formed in less than 100 µs (2). Since it is currently notpossible to resolve these length and time scales experimentally, theoretical orcomputational models are employed to gain insight into the process of fusionpore formation.

Theoretical descriptions are based on elastic theories for membrane sheets,which postulate intermediate configurations and try to find the lowest energytransition states. However, these lowest energy states usually correspond torelatively high energy barriers. In spite of modifications of the assumed in-termediates to lower the energy barriers, these barriers are still estimated tobe of the order of 40kBT .

Computer simulations such as Brownian Dynamics (3), Monte Carlo sim-ulations (4), coarse grained Molecular Dynamics (MD) (5–7), dissipative par-ticle dynamics (DPD) (8, 9), atomistic MD (10, 11), on the other hand, givea molecular picture of the process and are not restricted with respect to thestructure of intermediate configurations. These simulation studies observedifferent fusion pathways and highlight the importance of lipid conforma-tions in the process, but they do not usually allow one to measure the energybarriers between states. One exception is a Markovian state model based oncoarse grained MD (12, 13) that has managed to deduce the energy difference


between the initial state and several intermediates from the transition rates.In the present study, DPD simulations have been used to probe the statis-

tics of many fusion attempts, while still being able to simulate the relevantlength and time scales. From the statistics of the fusion time in combinationwith separate simulations of enforced interbilayer flips, in which one tail ofa lipid molecule is moved from one bilayer to the other leading to a splayedconformation of this lipid, the energy barriers for the fusion observed in thesesimulations could be estimated as already outlined in (14).

We focus on the presumably simplest way to induce lipid bilayer fusion,i.e. via a global membrane tension, which is coupled to hydrodynamics andcan be directly controlled in MD (15) and DPD (8, 16) simulations withexplicit water. The experimentally observed frequency of fusion events in-creases with osmotic inflation of the vesicles (17), which indicates that theenergy barriers for fusion can be reduced by increasing the membrane ten-sion. The simulations reported here attempt to elucidate the fusion pathwayof this mechanism and its dependence on specific lipid properties as well asthe membrane tension, and to reveal the energy barriers between the inter-mediate states. Unlike experimental suggestions, previous DPD simulations(8) did not indicate any tension dependent energy barriers. However, asshown in the present article, membranes built from the parameter set usedin (8) are characterized by fast exchange of lipids between adhering mem-branes. Investigation of the effects of individual parameters on the membraneproperties have allowed us to carefully adjust the simulation parameters to(i) obtain bilayers with improved stretching behavior and (ii) create bilayersthat exhibit an energy barrier to those interbilayer exchange of lipids. In thefollowing, we report a systematic variation of the simulation parameters anddiscuss the fusion pathway for the chosen parameter set with all intermediatestates. In particular, one of the simulation parameters that determines theenergy barrier of a relevant sub-process for fusion can be identified and thuscan be tuned by comparison with available data on this barrier.

Our fusion geometry consists of a vesicle with a diameter of 14 or 28nm in contact with a planar bilayer. To obtain sufficient statistics the timeevolution of over 160 fusion attempts of a vesicle to a planar bilayer patch ismonitored. In those simulations, the initial projected area per molecule, A,is varied systematically and serves as a control parameter.

Since we study tension-induced fusion of membranes and vesicles, a fewgeneral remarks about membrane tension are appropriate. At first sight,membrane tension seems to be analogous to the interfacial tension between


two fluid phases. The latter tension can be defined in two ways: (i) Ther-modynamically via an expansion of the system’s free energy in powers of thesystem size; and (ii) Mechanically via the work expended to increase the areaof the interface. This work can be expressed in terms of the pressure or stresstensor of the fluid system. Both definitions turn out to be equivalent eventhough this equivalence is far from obvious, see, e.g., (18).

Compared to interfaces, membranes have more configurational freedom.Indeed, in contrast to interfaces, membranes consist of thin molecular bi-layers that can form unilamellar or multilamellar vesicles and a variety ofthermodynamic phases. Furthermore, a single membrane can rupture, foldback on itself, or undergo other types of morphological transitions. In fact,a sufficiently large membrane segment that is stretched and, thus, undermechanical tension can always lower its free energy by rupture or poration.Therefore, a thermodynamic definition of membrane tension, which neces-sarily involves the limit of large membrane area, is beset with conceptualdifficulties. On the other hand, the mechanical definition of tension via thestress tensor can also be applied to relatively small membrane segments asstudied in computer simulations (15). Thus, when we use the term ‘mem-brane tension’, we always mean the mechanically defined tension.

The paper is organized as follows. In the Methods section, the simulationmethod and the model systems are summarized. Then follows a descriptionof the material properties of the simulated membranes, which emphasizes theimproved stretching behavior of the membranes and introduces a simplifiedimplementation of the tension measurements. The dependence of these prop-erties and the stretching behavior on the individual simulation parametersis discussed in the ’Parameter Dependence’ section. The following two sec-tions, ’The Fusion Pathway’ and ’Other Pathways and Fusion Statistics’, givea detailed description of possible time evolutions and outcomes. The inter-mediate stages of the fusion process are analyzed in detail and the statisticsof these are discussed. Finally, in ’Fusion Statistics and Energy Barriers’,the fusion time distributions and their tension dependence are analyzed toestimate the tension dependent energy barriers of the fusion pathway.


Methods

Simulation Method and Parameters

Dissipative particle dynamics (DPD) is a coarse grained, particle-based simu-lation technique that explicitly includes water and reproduces hydrodynamicbehavior (19, 20). The DPD particles or beads represent small volumes offluid rather than single atoms so that their interactions are softly repulsiveand short ranged, see Supplementary Material, Data S1, for details. All in-teraction potentials have the same range r0, but their amplitudes aij differfor different bead species. Both self-assembly and phase behavior of lipidshave been reproduced with DPD (16, 21, 22).

The systems considered here are built up from three bead species: lipidhead (H), lipid chain (C) and water (W) beads. The more complex archi-tecture of the lipid molecules, is constructed by connecting adjacent beadswith spring potentials. In addition, the hydrocarbon chains are stiffened bya bending potential for two consecutive bonds.

The model lipids have a headgroup consisting of three H beads and twohydrophobic tails, each of which is made up from four C beads, as shownin Figure 1. This architecture was introduced in the context of MolecularDynamics simulations (15) and also used in previous DPD simulations (8, 16).

In general, the simulation parameters are chosen to reproduce the meso-scopic behavior of the system. Thus, the overlap and interaction strength ofthe water beads used here reproduce the compressibility and density fluctu-ations of water. The remaining force amplitudes aij are fine-tuned to matchthe properties of lipid bilayers, by carefully determining the effects of changesto each parameter on the bilayer properties. There are some important con-straints for a reasonable bilayer model such as a well defined bilayer structurewithout interdigitation of the two monolayers, lateral fluidity and bendingflexibility. In addition, there are inherent limits to the range of reasonablevalues of aij (23). Our choice of the force parameters also reproduces the bi-layer thickness relative to the molecular area A/N, the area expansion mod-ulus and, in addition, a reasonable barrier against lipid exchange betweencontacting bilayers.

This strategy to obtain the simulation parameters by adjusting them toyield the correct mesoscopic behavior is rather similar to the one used toobtain appropriate interaction parameters in atomistic simulations. Atom-istic force fields must also be optimized for different situations to reproduce


experimental results, as demonstrated e.g. by the large number of atomisticwater models, see (24).

As the simulated membranes in our model are relatively thin compared tothe area per molecule, DMPC, which is a common, membrane forming lipidwith a comparatively short chain length of C14, is used as reference moleculefor comparisons with experimental data. To represent DMPC, one tail beadmust correspond to 3.5 CH2 groups.

(a) aij H C WH 25 50 35C 50 25 75W 35 75 25

(b) aij H C WH 30 35 30C 35 10 75W 30 75 25

Table 1: (a) Old parameter set for the conservative force in DPD. All aii val-ues are chosen to reproduce the compressibility of water. (b) New parameterset as used here. All values of aij satisfy aij ≥ 10 to ensure correct diffusivebehavior (23)

The DPD simulations reported here use the set of force amplitudes aij

as given in Table 1(b). These represent an improved set compared to thosein Table 1(a) used in previous simulations (8, 16). The tail chains have abending stiffness k3 = 15 compared to k3 = 20 used before but all otherparameters are the same as in (8). The new set of force amplitudes improvesthe properties of the resulting membranes in two respects: (i) in their overallstretching behavior (see Results Section, Stretching Behavior) and (ii) in thestability of two adhering membranes against lipid exchange between thosemembranes. The previous parameter set in Table 1(a) shows no observableenergy barrier to interbilayer exchange of lipids in adhering bilayers, makingthis state highly unstable. Lipid exchange begins immediately and the twobilayers intermix. This does not reflect a realistic situation, as hydration ofthe lipid head groups will present a considerable barrier for the hydrophobictails and have a stabilizing effect on adhesion. Note that all interactionstrengths aij satisfy aij ≥ 10, ensuring correct diffusive behavior (23).

The large value of the chain water interaction, aCW = 75, was kept fromthe original parameter set. It leads to the strong effective adhesion betweenmembranes that come into close proximity. Presumably, it will also lead to


a relatively large hydration energy of the lipids. However, since the fusionpathway discussed below does not involve direct contact of chain and waterbeads, this overestimation of the hydration energy is not expected to changethe results.

The head tail interaction, aHT, is optimized to reproduce the energy bar-rier presented by the hydrated headgroups against interbilayer flips. On theother hand, this interaction is not optimized with respect to flip-flops withinone bilayer, which occur more frequently than in experimental membranes.In simulations of a single bilayer containing 1640 lipids, 12 flip-flops from onemonolayer to the other are observed within one microsecond. This would cor-respond to an average dwell time of 1 ms of a lipid in one monolayer. Thisis faster than observed experimentally, where the dwell time is on the orderof 1 h (25, 26).

Another set of force parameters has been used in Ref. (27). The latter setuses different force amplitudes aij (i) for pairs of C beads that belong to thesame or to different chains, and (ii) for the end beads Ce of the chains, whichleads to even more parameters that need to be adjusted. In contrast, for theparameter set used here all pairs of C beads are governed by the same forceamplitudes as given in Table 1(b). In (27) the repulsion between differenttail chains is much smaller than between the beads along one chain, whichintroduces an effective attraction between tail chains. The resulting bilayersare much stiffer than those in our model and the tail chains are much moreordered and aligned. This leads to a different fusion pathway as discussed atthe end of the subsection ‘Energy Barrier for Interbilayer Flips ”.

Implementation of the Stress Tensor

The calculation of the mechanical membrane tension Σ in the simulations isbased on the macroscopic stress tensor Σαβ. From its components the stressprofile s(z) = 0.5(Σxx + Σyy)−Σzz can be calculated with the method intro-duced by Schofield and Henderson (28) and extended to m-body potentialsby Goetz and Lipowsky (15). The membrane tension Σ is the z-integral overthe stress profile s(z).

To obtain the stress profile s(z) in a simulation, the simulation box issubdivided into thin slices and the slice integration is expressed by multiply-ing the expressions with two Heaviside step functions as described in (15).The membrane tension, on the other hand, depends only on the z integratedcomponents of this averaged stress tensor. It can therefore be calculated by


directly integrating the components of the stress tensor in the z direction.This integration avoids the use of step functions and leads to expressionsthat are easy to implement.

In the calculations, the contributions to the microscopic stress tensorσαβ are separated into contributions from interaction clusters of size m, i.e.into two-body, three-body etc. interactions with corresponding interactionpotentials U (2), U (3) etc, as described in the Supplementary Material, DataS2. For the model considered here, only two-body interactions from theconservative interaction and the bond potentials between two beads, andthree-body interactions from the bondpair potentials contribute to the stresstensor Σαβ. The z-integrated contributions from the two-body interactions,Iαβ2 , and from the three-body interactions, Iαβ

3 , are given by

Iαβ2 =

1

L2‖

∑

〈k,l〉

( ∂

∂rαjkjl

)

U (2)(rjkjl) rβ

jkjl(1)

and

Iαβ3 =

1

3L2‖

(2)

×∑

〈j〉

∑

〈k,l〉

[

(

∂∂rα

jl

)

U (3)({r}) −(

∂∂rα

jk

)

U (3)({r})]

rβjkjl,

where L‖ is the lateral size of the membrane and the superscripts α and βcan take the values x, y or z and specify the respective vector components.The sum over 〈j〉 is over all m-clusters, in this case over all triplets, and 〈k, l〉denotes all particle pairs within the cluster with positions rjk

and rjl, so that

rjkjlis the vector connecting the two particles.

These expressions correspond to the generalized 2nd and 3rd virials. Thesum over the contributions from Iαβ

2 and Iαβ3 give the αβ-components of

the z-integrated total stress tensor. Using α = β = x and α = β = zthe tangential components of the macroscopic stress tensor, Σxx = (Ixx

2 +Ixx3 ) and Σyy = Iyy

2 + Iyy3 , and the normal component Σzz = (Izz

2 + Izz3 ),

which define the membrane tension, are found. As the membrane is fluid,all components with α 6= β should vanish. This is fulfilled in the simulationswithin ±0.002 kBT/r

30. The tension calculated from this direct integration

is plotted in Figure 2 together with the tension calculated from the detailedstress profile. Clearly, the two methods give the same result.


Rescaled Parameters in Dissipative Particle Dynamics

The bead diameter r0 and bead mass m0 represent natural units of lengthand mass in the simulation. Energies are measured in units of kBT . Thebasic unit of time constructed from these quantities is τ =

√

r20 m0/kBT .

It is convenient for the analysis of the simulation results to introduce di-mensionless quantities, which will be indicated by a bar. Thus, we definethe dimensionless area per molecule A ≡ A/r2

0 and the dimensionless tensionΣ ≡ Σr2

0/kBT with corresponding compressibility modulus KA ≡ KAr20/kBT .

To obtain physical units, the length and time scale r0 and τ of the systemmust be chosen in an appropriate manner from characteristic properties of thesystem. For studies of fluid membranes, the area per molecule A of a tensionfree bilayer and the diffusion coefficient D‖ for lateral diffusion of lipids inthe bilayer provide natural length and time scales. The area per moleculein the simulations is 1.25 r2

0, whereas the experimentally measured value is0.596 nm2 (29), so that the basic lengthscale is r0 = 0.69nm. The inplanediffusion coefficient of DMPC is approximately D‖ = 5 µm2/s (30). In thesimulations, the inplane diffusion coefficient of the lipids is D‖ = 0.016r2

0/τ .Thus, for the correct diffusive behavior a mapping of τ = 1.6 ns is required.Accordingly, a ∆t = 0.02 τ timestep in the simulations corresponds to ∆t =0.0314 ns.

The Simulated Systems

All simulations are done in the NVT ensemble with a density ρ = 3/r30.

The choice of NVT or NPT depends on the system that is simulated. Ifthe membranes represent a small piece of a much larger membrane a smallfusion site with a vesicle will have little effect on the membrane tension andNPT would be an appropriate ensemble. For two membranes of roughly thesame size a pore or fusion site will reduce the membrane tension considerablyand the effects are better represented by an NVT ensemble. The membranesin our simulations reach experimentally feasible sizes and the NVT ensemblehas the advantage that the results are comparable to simulations of two fusingvesicles with the same diameter, which behave very similar to the fusion toa planar bilayer described here.

In this ensemble the projected membrane area remains constant, whereasthe bilayer tension fluctuates around its average value. The standard devia-tion of the tension values decreases strongly with increasing system size L‖


up to L‖ ≃ 30r0, and then seems to approach a constant value, see Fig. S1in the supplementary material.

In the present study, four types of simulations have been performed (seeSupplementary Material, Data S3): (i) a random mixture of lipids and solventto test whether the lipids self-assemble to a bilayer structure, (ii) a pre-assembled planar bilayer for characterization of the membrane properties,(iii) a vesicle in close proximity to a planar bilayer patch to observe fusion,and (iv) two adhering planar bilayers, which are used to measure the averagework required to enforce the flipping of one lipid tails from one bilayer to theother.

Bilayers and vesicles are assembled with a prescribed area per molecule,A. The planar bilayer is built up from 6700-8300 lipid molecules, the 28 nmvesicle from 6800-7500 lipids, and the 14 nm vesicle from 1400-1600 lipids.The membranes are equilibrated with a smaller timestep of ∆t = 0.005 τ for10 ns. For fusion simulations, a vesicle is placed in close proximity to a planarbilayer patch, separated by a thin layer of water beads. The thickness of thisinitial water layer is about 1.5 nm. For such a separation,the two membranesusually come into contact by diffusion within less than 150 ns. No externalforces are applied to bring the membranes into contact. Those simulationsfor which the vesicle diffuses away from the planar bilayer are discarded.

Fusion is induced by applying a lateral tension to the planar bilayer.This is achieved by increasing the value of A, which is directly related tothe bilayer tension. Most fusion simulations have been done for two differentvesicle diameters, 20 r0 and 40 r0, corresponding to approximately 14 and 28nm, respectively, in a simulation box with an area of (72 r0)

2 ≃ (50 nm)2,and a height of 52 r0 and 72 r0 for the 20 r0 and the 40 r0 vesicle, respectively.Additional simulations of a vesicle with diameter 20 r0 in a smaller box of(36 r0)

3 were performed to further explore the observed dependence of theresults on the vesicle size. To obtain relevant fusion statistics, more than 160fusion simulations have been monitored. For each data point correspondingto a particular value of the control parameter, an average over at least 18independent simulations is taken.

To measure the flipping energy, the average work required for an inter-bilayer flip, a single lipid from the lower bilayer is selected and a slowlymoving, external harmonic potential is applied to one of its tail beads as inFigure 3(a). The potential starts centered on the bead’s z-coordinate andmoves slowly towards the other bilayer at a constant speed of 0.009 nm/ns,until the tail has flipped into the other bilayer and the lipid has assumed


an splayed conformation as in figure 3 (b). At each timestep, both the dis-placement of the bead from its original position, zbead, and the spring forcerequired to hold it at that position are recorded.

To simulate quasi-static pulling, the motion has to be sufficiently slow.The appropriate velocity can be determined from simulations pulling a lipidthrough the flat energy landscape of a uniform solvent. Fast pulling leadsto a non-zero spring force, caused by the friction of the surrounding fluid.The friction coefficient found in these simulations is approximately 2× 10−10

ns/m. At sufficiently low pulling speeds, the average position of the bead isalways close to the position of the harmonic potential.

Unlike steered MD simulations that usually pull along a given directionvector, which can introduce a bias if the system cannot adapt its orientationon the MD timescale, in our simulations, the potential is applied only to thez coordinate of the bead, which is free to move in the xy plane. Furthermore,the low pulling speed and small size of the molecule make it unlikely thatthe direction will bias the resulting work, as molecules can easily adapt.

The simulation code used for these simulations has been developed andtested by our group. More information about this simulation code may beobtained from the authors. Simulations were run on single processors (IntelXenon 3.6GHz), and needed about 7 days/µs

Results and Discussion

Material Properties of the Simulated Bilayers

Simulations starting from a random mixture of lipids and solvent show, thatthe molecules with a H3(C4)2 architecture as shown in Figure 1 and theinteraction parameter set in Table 1 (b) self-assemble into bilayer vesicles(Supplementary Material, Fig. S2). With the length and time scale of thesystem determined from the area per molecule A of the tension free bilayerand inplane diffusion coefficient D‖, other equilibrium properties of the bi-layer can be measured and compared to experimental findings.

The density profiles of the individual components show that the lipidsform proper bilayer structures with water completely excluded from the inte-rior with the head beads accumulated at the interface between hydrophobictails and water (Supplementary Material, Fig. S3). The two monolayers arewell separated and there is no interdigitation. The weak repulsion aCC = 10


leads to relatively narrow bilayers with a high density of the hydrophobiccore. As a consequence, when mapping the amphiphiles to real molecules,a lower mass is associated with the tail beads as demonstrated by V. Ortizet al. (31). When the bilayers are stretched, the volume per lipid remainsapproximately constant as is observed experimentally for the incompressiblehydrocarbon chains.

Stretching Behavior

In the fusion simulations, fusion is induced by controlling the area per moleculein the bilayers, which effectively puts the membranes under tension. To studytension dependent behavior, a knowledge of how the tension Σ, in the bilayerdepends on the molecular area is required.

The stress profile across the membrane has similar characteristics to thosefound in other coarse grained simulations (15, 16), with positive peaks frominteractions with water at the lipid-solvent interface and negative contri-butions from the bond and bondangle potentials in the bilayer center, seeSupplementary Material, Fig. S4.

The membrane tension Σ is the z-integral over the stress profile s(z). Tofind the dependence of Σ on the molecular area, a series of simulations atdifferent values of A is performed. Figure 2 shows the membrane tension Σ asa function of the area per molecule A for the parameter set introduced here incomparison to the previously used set. The comparison shows that whereasthe area per molecule of the tensionless state is the same, A0 = 1.25, in bothcases, the stretching behavior of the two membranes is rather different. Theold parameter set leads to very stable membranes, whose area per moleculecan be increased by about 60 % without rupturing the membrane within afew µs. The dimensionless tension Σ ≡ Σr2

0/kBT is a nonlinear function ofthis molecular area, increasing steeply from the tensionless state but thengrowing much more slowly. Experimental studies of large vesicles show thatthese cannot sustain extensions of more than 3 to 5 % for lipid vesicles (32)and about 20 % for polymersomes (33). The new parameter set reducesthe stretchability of the simulated membrane to about 20 % and leads toan essentially linear relation between tension Σ and molecular area A, asis also observed experimentally. The very large stretchability is a commonproperty of simulated bilayers and should be attributed to the small lengthand time scales. For example, at the limit of metastability for the tenseplanar membrane, corresponding to the maximum tension value in Figure 2,


the bilayer remains stable over the 1.5 µs simulation performed to obtain thisdata, whereas a pore forms in a longer simulation of 9 µs. In addition, thisstability limit for the stretched membrane should also depend on the boxsize. In Ref. (34) the free energies of a stretched and a porated membraneboth with the same base area and containing the same number of molecules,have been compared. The analysis leads to a stability limit of the poratedmembrane, and a coexistence point for the two states. The correspondingstretch (A− A0)/A0 scales as L

−2/3‖ with the linear size L‖ of the base area.

The stability limit of the stretched membrane on the other hand lies atinfinite stretch. If we assumed that these scaling laws can also be appliedto the rupture of a membrane, a 100 µm2 DPD membrane would only bestretchable by 0.7 %.

The linear relation between Σ and A implies that the molecular area,which is used as a control parameter, can easily be converted to membranetension in order to describe tension dependent processes. The area compress-ibility modulus KA is given by the slope of the tension as a function of Aat the tensionless state with A0. The corresponding dimensionless modulusKA ≡ KAr

20/kBT can be deduced directly from the plot in Figure 2 and has

the value KA = 22.75. Converted back to physical units, this leads to a valueof 200 dyn/cm, which is slightly lower, but of the same order of magnitude asthe experimentally measured values for phospholipids, which range from 234dyn/cm for DMPC to 265 dyn/cm for DOPC (29). Thus the agreement withexperiment is greatly improved compared to the previous DPD parameterset which leads to KA = 700 − 1000 dyn/cm.

Parameter Dependence of Bilayer Equilibrium Proper-

ties

To improve the macroscopic properties of the simulated bilayers, the variousDPD parameters have been varied systematically. In this study, the stretch-ing behavior, i.e. the membrane tension Σ as a function of molecular area A,is of particular interest, because membrane tension is used to induce fusion.

Here we give a brief overview of the effects of different parameter changes.Their effects on the membrane’s stretching behavior are summarized in theSupplementary Material, Fig. S5(a).

Changing the relative magnitudes of the head-head and tail-tail forceamplitudes, aHH and aCC, changes the effective size of the head groups or tail


chains, respectively, and alters their compressibility and thus magnitudes inthe density profile: lower aii values lead to higher densities and vice versa.If the values become too large or too small the spontaneous curvature ofthe monolayers becomes too large and the bilayer state becomes unstable.These effective size changes of the beads also affect the membrane stretchingbehavior. Larger head groups, corresponding to larger aHH values, shield thehydrophobic tails more effectively from water, so that the bilayer becomesmore stable. The functional dependence of membrane tension Σ on moleculararea becomes approximately linear and the molecular area of the tension freebilayer increases, as the larger head groups need more space. Smaller valuesof aCC, on the other hand, corresponding to smaller, more compressible tailchains, decrease the membrane stretchability and also lead to more lineartension curves. In the final parameter set of Table 1(b), the combinationof the relatively strong head bead interactions and the closer packing of thechains due to their reduced repulsion, generate a monolayer curvature thatfacilitates lipid rearrangements to cover the rim of a forming pore.

The tail water force amplitude aCW represents the hydrophobic effect.It has to be strong enough to drive self assembly, but otherwise has littleeffect on the tension’s dependence on the molecular area. A slight increasein the values of Σ appears as the tail water contributions to the stress profileincrease, but this effect is small if the hydrophobic region is well shielded bythe head groups.

The effect of chain length on the membrane properties has been subjectto both experimental and simulation studies (16, 33, 35, 36). The simula-tion studies have found that with longer tails, the packing of lipids becomesdenser and that both the area stretch modulus KA and the tensionless valueof the reduced molecular area A depend sensitively on the chain length. Theexperiments, on the other hand, found that the area stretch modulus is in-dependent of chain length over a wide range (from 13 to 22 carbons/chain).The simulations performed here in the context of the parameter optimiza-tion show that the chainlength dependence of the bilayer behavior varieswith the parameter set used in the simulations. Figure 4(a) and (b) showthe membrane tension Σ as a function of reduced molecular area A for lipidswith different chain lengths for the two parameter sets in Table 1 (a) and(b). In both cases, the stability of the tense membrane increases with chainlength. However, in the original parameter set (Table 1(a)), the ‘bump’ inthe tension curve becomes more pronounced as the chain length increases,dividing the stretching process into two regimes with a high and a low ten-


sion gradient. At the same time, the compressibility modulus KA increaseswith chainlength. For the new parameter set used here as given in Table1(b), the curves also start to show a slight deviation from the linear depen-dence. In contrast to the original parameter set, the compressibility modulusnow decreases slightly with chainlength. However, both effects are much lesspronounced than in the original case, with a deviation in the compressibilitymodulus KA of about 1% for an increase in chain length from three to fivebeads.

Finally, the parameter ratio aHC/aCC, also plays an important role. Thisratio represents a measure of how strongly headgroups and hydrocarbonchains repel each other. The old parameter set is characterized by aHC/aCC =2, which does not suffice to stabilize two adhering bilayers. In the lattersituation, a relatively large number of lipid tails starts immediately to un-dergo interbilayer flips, leading to a strongly perturbed adhesion zone on thetimescale of 1 µs. If the ratio aHC/aCC is increased, interbilayer flips becomenoticeably slower. For the larger ratio of aHC/aCC = 3.5 as used here, thebarrier for interbilayer flips is measured to be approximately 8 kBT .

The Fusion Pathway

To shed light on the molecular mechanism of membrane fusion, more than160 DPD simulations of fusion attempts between a vesicle and a tense planarbilayer patch have been monitored. Successful fusion attempts in these sim-ulations all involve the same sequence of events. Here we present a detaileddescription of the observed fusion mechanism.

The simulations start with the vesicle in close proximity to the planarbilayer. The evolution of the system is monitored from the first contactbetween vesicle and planar bilayer until the fusion pore has opened. If fusionhas not occurred within 20µs the attempt is counted as unsuccessful.

Figure 5 shows snapshots of the development of one successful fusionevent between a planar bilayer, (red heads/green tails) and a vesicle (orangeheads, yellow tails) with a diameter of 30 nm. The system is shown from twoperspectives, where the z axis is taken to be normal to the planar bilayer:cross sections viewed from the side and cuts through the planar membrane,viewed from above. For the topviews, the cuts are performed through themidplane of the planar membrane. In these topviews, all the green tail beads,which originally belonged to the planar membrane, are made transparent. Inthe upper topviews, one sees the appearance of the yellow tail beads that


enter the planar membrane by interbilayer flips from the vesicle. In thelower topviews, the yellow tailbeads are transparent as well, revealing theorange head beads of the vesicle lipids. The white areas in the lower topviewscorrespond to hydrophobic regions containing only tailbeads. A few waterparticles can be seen outside the vesicle. These have leaked through themembrane during the initial equilibration period, where the lipids are linearand arranged on a lattice. There is no leakage at later times.

Upon first contact, the vesicle adheres to the planar membrane patch.The contact area grows and the vesicle membrane spreads onto the planarmembrane, forming a relatively sharp contact angle at the contact line, i.e.the boundary of the contact area. As the contact area grows, lipid tails startto move from the vesicle into the planar bilayer. These exchange processes orinterbilayer flips take place mainly at the contact line, see Fig. 5(b), because(i) there the vesicle membrane has a pronounced ‘kink’ of high curvature withincreased strain and more strongly compressed lipid tails, a situation thathas been previously discussed in another context (37), and (ii) the vesiclelipids are already tilted with respect to the planar bilayer. Both effects tendto lower the energy cost of interbilayer flips along the contact line.

The interbilayer flipping of the lipid tails disturbs the local double-bilayerstructure and leads to the formation of a disordered membrane domain withinthe contact zone. The hydrophobic tails moving through the headgroupsbring the hydrophobic centers of the two bilayers into direct contact. Thishydrophobic contact expands not radially symmetric, but rather followingthe contact line, assuming a bean-like shape, which can be seen as the whiteareas in Figure 5(b).

Finally, within this disordered hydrophobic-contact region, lipids reorderto form a small patch of a single, hemifused bilayer with a diameter of afew nanometers. The hemifused bilayer leads to a more favorable area permolecule and thus reduces the bilayer tension. It expands for a short timeand finally ruptures at the rim to form the fusion pore. In the exampleshown in Figure 5, this takes place 1334 ns after the onset of the process.All successful fusion events observed in our simulations involve the samesequence of membrane conformations described here.

In addition, several fusion events between two vesicles with a diameter of28 nm have been simulated. The fusion of the two vesicles follows the samepathway as the fusion between a vesicle and a planar membrane.


Adhesion and Interbilayer Flips

In our DPD simulations, the adhesion of bilayers arises because of two effec-tively attractive interactions. First, the repulsive interaction aWC betweenthe water beads and the chain beads is larger than the repulsive interactionaHC between the head beads of one membrane and the chain beads of an-other membrane. Therefore, the interaction energy is reduced if the waterbeads adjacent to one membrane are replaced by another membrane. Sec-ond, the small water beads push the large membranes together by depletioninteractions of entropic origin (38).

Experimentally, the adhesion of DMPC membranes has been somewhatcontroversial. Multilayer stacks of DMPC membranes in excess water exhibitan equilibrium spacing of about 2.8 nm (39). Such stacks have also been ob-served to form spontaneously at the air-water interface (40). When twoDMPC membranes are immobilized on mica surfaces, their adhesion energyW was estimated to be W ≃ 0.1 mJ/m2 as deduced from surface force mea-surements (41). On the other hand, the adhesion of DMPC vesicles as studiedby micropipette aspiration led to the smaller estimate W ≃ 0.01 mJ/m2 (42).Furthermore, the group of Helfrich reported that large DMPC membranesdo not adhere in distilled water unless they experience some tension (43).

These different observations can be reconciled by the following theoreticalconsiderations (44). The van der Waals interaction between two planar mem-branes that have the same lipid composition is always attractive as followsfrom the general Lifshitz theory for these forces, see, e.g. (45) and referencestherein. This van der Waals interaction is, however, renormalized by theshape fluctuations or undulations of the membranes provided the membranetension is sufficiently low (44, 46). This effect is stronger for larger membranesegments, since the number of undulation modes is proportional to the mem-brane area. Thus, for sufficiently large membrane segments, the undulationscan lead to a renormalized interaction that is purely repulsive.

In the surface force experiments, all membrane undulations will be sup-pressed and one should, thus, measure the bare van der Waals attraction notaffected by undulations. In the micropipette experiments, membrane undu-lations are also suppressed but only down to a certain minimal wavelengththat depends on the applied tension (44). The remaining fluctuations still actto reduce the effective van der Waals attraction which explains the smallervalue of the adhesion energy as estimated from the micropipette experiments.

In the system considered here, membrane undulations are suppressed for


two different reasons. First, because of the small size of the vesicles, thenumber of possible undulation modes is rather limited even in the absence oftension. Second, these remaining undulation modes are further suppressed bythe tension Σ. Therefore, the planar membrane and the vesicle should exhibitan adhesion energy of the order of W ≃ 0.1 mJ/m2 or 2.5×10−2 kBT/nm2 atroom temperature. As shown in the following, the DPD membranes studiedhere have an adhesion energy which is rather similar to this experimentallydetermined value even though the DPD parameters were not optimized withrespect to this energy.

In our simulations, the system remains in the adhered state for a long timeif the initial membrane tension is relatively small. The contact area growsuntil the system reaches a mechanically stable state, in which the energygain from the effective adhesion energy density |W | is balanced by the costof deforming the membranes.

Because the membranes attain a relatively large tension during the ad-hesion, the membrane shapes are tension dominated and approach sphericalcaps. The contact curvature radius Rco observed in the snapshots is smallcompared to the linear size of the vesicle Rve . Thus as Rco = (κc/2|W |) ≪Rve, the bending energy is negligible and an effective contact angle θeff canbe defined as in (47). These simple system geometries can be mechanicallystable and thus yield estimates of further mechanical properties of the systemfrom simulation snapshots.

A balance of the forces on the contact line in the directions parallel andperpendicular to the planar bilayer leads to two independent Neumann equa-tions for the tensions in the planar bilayer, Σpl and the vesicle membrane,Σve (Supplementary Material, Data S4)

(Σpl + Σve − |W |) cos(θ1) + Σve cos(θ2) − Σpl = 0

(Σpl + Σve − |W |) sin(θ1) − Σve sin(θ2) = 0(3)

where θ1 and θ2 are the angles formed by the membrane segments of thevesicle with the bilayer plane as sketched in Figure 6.

The geometry of the adhering membranes consists of three membranesegments: two spherical cap regions for the vesicle and the contact area,and a planar segment. For this geometry, the area per lipid in the planarmembrane can be calculated and the corresponding tension Σpl determined


from the tension-area plot in Figure 2. If this tension value is inserted intothe Neumann equations 3, these two equations can be solved for |W | and Σve.In Figure 6(b) the result for the average adhesion energy |W | is displayed asa function of the initial molecular area in the planar membrane. Inspectionof this figure shows that |W | is linearly related to A with the best fit givenby

|W | r20/kBT ≃ 0.01 + 6.012(A− A0). (4)

For A = A0, the strength of the attractive interaction between the membranesis thus about 2× 10−2kBT/nm

2, which is of the same order of magnitude asthe values of the van der Waals attraction between PC membranes obtainedfrom surface force apparatus measurements, see e.g. (41). The tensionlessmolecular area of the vesicle is found to be (A0)ve ≃ 1.05r2

0. These estimatesrepresent a lower bound for |W | and A0, as, especially at low tensions, otherfactors such as the bending energy may also contribute to the force balance.

This mechanically stable adhered state is characterized by a low rate ofinterbilayer flips. Figure 7(a) illustrates such interbilayer flips at low tension.The two images are constructed in the same way as the upper top views inFigure 5: All beads of the planar membrane except the headbeads of theproximal monolayer are made transparent. Thus one sees the appearanceof the yellow tail beads that enter the planar membrane by interbilayer flipsfrom the vesicle. Inspection of Figure 7(a) reveals that these interbilayer flipspreferentially occur along the circular contact line. In addition, this figureshows an example for inplane lipid diffusion, as indicated by the green arrow:after both tails of a lipid molecule have undergone an interbilayer flip, thelipid molecule is free to diffuse away from the contact area.

A closer look at individual interbilayer flips is shown in Figure 7(b), whichdisplays a view inside the planar bilayer, with its hydrophobic beads madetransparent, so that one can observe how the first vesicle lipid chain (yellow)moves over into the planar bilayer core. In the vicinity of lipids in this splayedtail conformation, the probability of further interbilayer flips is somewhatincreased.

A similar onset of fusion with splayed-tail lipids at the rim of the contactarea was also reported for other simulation studies (6). In another case,splayed lipids bridge the gap between the vesicles with a stalk forming aroundthem (7). Of course the splayed lipid conformation is a mechanism that canonly be observed in simulations using double tailed amphiphile models.

There is also some experimental support for a splayed tail configuration


at the onset of fusion, summarized in (48). Furthermore it has been arguedthat the most likely effect of the fusion peptide hemagglutinin (HA) is topromote inter-membrane flips by dehydration and structural disruption aspeptides replace water molecules in the hydration layers (1). Finally it isinteresting that, although mediated by fusion proteins, the fusion of vacuolesalso takes place at the edge of an extended contact area (49).

The fusion pathway described here provides a direct connection betweenthe onset of fusion and splayed lipid conformations. The latter conformationsare more likely to occur for cone-like lipids, i.e., for lipid molecules thatresemble truncated cones with a relatively small head group and relativelybulky tails (sometimes called lipids with ‘negative spontaneous curvature’).Indeed, when such a cone-like lipid is located in the proximal monolayer ofthe membrane kink along the contact line, its tails are strongly compressed,and the molecule can relax this mechanical strain by flipping one tail intothe proximal monolayer of the other membrane. Therefore, according to oursimulation results, cone-like lipids should enhance the fusion process becausetheir tails are more likely to undergo interbilayer flips and these lipids aremore likely to attain splayed conformations. In this way, we provide a newinterpretation to experimental observations as reviewed in (50, 51) that cone-like lipids act to promote membrane fusion.

Hydrophobic Contact-Disorder at the Contact Line

Hydrophobic contact occurs if several vesicle lipids flip to the planar bi-layer simultaneously and in close proximity. This strongly perturbs the localbilayer structure, where the head groups are forced apart and brings thehydrophobic cores of the two bilayers into direct contact, so that the tailchains intermix, as in the example in Figure 8(a). The combined width ofthe two membranes becomes thinner, since the two intervening layers of headgroups are no longer present. Often small groups of lipid heads remain attheir original location, i.e. between the two bilayers. As their tail chainsrearrange to form the hydrophobic contact, the head groups become trappedwithin this extended hydrophobic region, as they cannot easily move acrossthe hydrophobic material to either side. These sometimes appear later inlarger numbers in the distal monolayer of the planar bilayer. Trapped head-groups are also sometimes observed in atomistic simulations (11) and someexperimental studies (52, 53) observe fast movement of headgroups to thedistal monolayer during fusion. An example of this phenomenon is shown in


Figure 8(b)

The Hemifused Diaphragm and Formation of the Fusion Pore

After some time, a newly ordered hemifused bilayer forms within the per-turbed membrane region and expands until it either ruptures or the area permolecule is sufficiently relaxed. If fusion is successful, the size of the hemi-fused patch usually does not exceed a couple of nanometers and the timeuntil rupture lies between 100 ns and 300 ns. However, as the hemifusedpatch expands, the membranes gain additional area and the tension relaxes.Therefore, beyond a certain size hemifusion is stable on the timescale of thesimulations. In that case, the expansion of the hemifused area continues ei-ther until the tension is balanced or until it spans the entire contact area, inwhich case some tension may remain.

This extended hemifused state is often accompanied by some leakage ofsolvent beads through small transient pores that form at the contact line,as seen in Figure 9. The formation of such pores indicates that their linetension is strongly reduced, and confirms that the fusion pore or neck formsat the rim of the hemifused patch. Via these pores, mismatched values of Ain the inner and outer monolayer can also be equilibrated.

A geometric analysis similar to that of the adhered state gives an estimateof the net area per molecule (Supplementary Material, Data S4). In thecases where the hemifused patch covers only part of the contact area theseestimates are very close to the value of A for which the membrane is relaxed.If the contact area is completely hemifused, the tension in all three membranesegments should be balanced. Thus if the tension is not eliminated, the threecontact angles should be equal, unless there is a line tension to reduce thelength of the three membrane junction line. A study of snapshots of suchhemifused systems, where some membrane tension remains shows that in factthe internal angle of the vesicle is larger. This leads to an estimate of about2 kBT/r0 for the line tension as explained in the Supplementary Material(S4).

Other Pathways and fusion statistics

As the (meta) stable adhesion and hemifusion show, fusion is not the onlymechanism of relaxing the tension. Alternatively, tense membranes can rup-ture or, at low A, simulations may still remain in adhesion or hemifusion


after 20 µs. Unlike adhesion and hemifusion, bilayer rupture implies a def-inite failure of the fusion process. For tensions close to the rupture pointA = 1.6, bilayer rupture can easily be induced by the area increase due tothe deformations caused by the adhering vesicle. This process competes withthe fusion process and may thus happen before fusion takes place.

Comparison of the two vesicle sizes shows that (i) the small vesicle canfuse at lower values of A than the larger one and (ii) at low tensions, thelarge vesicle remains in the adhered state, whereas the small vesicle typi-cally hemifuses. These differences are related to the relative size of the twomembranes. To have a similar effect on the tension of the planar bilayerthe smaller vesicle has to be more strongly perturbed. Using the adhesionstrength obtained from the spherical fits and solving the force balance forstable adhesion, Eq. 3, for the shape of the adhering membranes, with theconstraint that the volume of solvent inside the vesicle remains constant, onefinds that the tension difference induced by the area increase of the deforma-tion is small compared to the overall tension in the membrane, so that bothvesicles would form approximately the same angles with the planar bilayerat a given A. As a result the contact area of the 28 nm vesicle is larger bya factor (Rl/Rs)

2 = 4, covering ∼ 20% of the planar bilayer, whereas the 14nm vesicle can cover only ∼ 5%. As the adhesion energy is proportional tothe adhesion area, the energy reduction for the small vesicle is not sufficientto achieve this at most tensions and the hemifused intermediate is formed,unless the vesicle is practically tension free. Then, however, the tension isnot high enough anymore to rupture the hemifused patch and form the fusionpore. As a consequence, unsuccessful fusion attempts of this smaller vesiclesize are usually trapped in the hemifused state, whereas larger vesicles at thesame tension often remain in the adhered state.

Fusion Statistics and Energy Barriers

Fusion time distribution

The tension determines not only the success rates, but also the time scaleof fusion. Each successful fusion event has a corresponding fusion time, tfu,defined as the time elapsed from first contact between the vesicle and theplanar membrane patch until the formation of the fusion pore has been com-pleted. The distribution of these fusion times depends on the membranetension, as already described in (14). Each fusion time distribution, corre-


sponding to a certain A, can be characterized by the average value 〈tfu〉 andthe width ∆tfu ≡

√

〈(tfu − 〈tfu〉a)2〉. Both the average fusion times obtainedfrom the fusion time distributions of the simulations, 〈tfu〉a and the widthof the distributions are shown as functions of the molecular area A in Fig-ure 10. Inspection of this figure shows that both quantities appear to growexponentially with decreasing (A). It becomes, therefore, increasingly diffi-cult to determine the time scale of fusion from computer simulations as thetensionless state is approached. In order to obtain an accurate estimate, notonly the longer average fusion times themselves have to be accessible to thesimulations, but they have to also sample the increasingly broad distribution.

The different outcomes and overlapping time distributions observed inthese simulations demonstrate clearly that the results of individual trajecto-ries should not be over-interpreted. To obtain reliable results, or quantitativerelations, such as the tension dependence of the fusion times, it is necessaryto perform a large number of runs.

Because these distributions only include successful fusion events and dis-regard final states of adhering or hemifused vesicles, which might still go onto fuse after longer waiting times, the average values obtained from thesedistributions represent lower bounds for the average fusion times. An upperbound can also be obtained from a second estimate that averages over the fu-sion rates 1

tfuand includes the adhering and hemifused final states as 1

tfu= 0.

The resulting upper bound of the average fusion time, 〈tfu〉b = 1〈1/tfu〉

, is also

presented in Figure 10. At high bilayer tensions, the data points for 〈tfu〉bmore or less coincide with those for 〈tfu〉a, but at low tensions, 〈tfu〉b devi-ates from 〈tfu〉a towards longer fusion times, and diverges when no successfulfusion events are observed. The true dependence of the average fusion time〈tfu〉 on the molecular area A lies between the two data sets for 〈tfu〉a and〈tfu〉b.

Overall Energy Barrier

The tension dependent fusion times indicate a tension dependent energybarrier for fusion. In an attempt to identify states that may constitute such abarrier, the fusion process has been decomposed into three sub-processes. (i)Sub-process α starts with the first contact between the two membranes andrepresents the adhesion and spreading of the vesicle onto the planar bilayerup to the time when the first interbilayer flip of a lipid tail has taken place.The duration of this process defines the first flipping time tα; (ii) Sub-process


β consists of the intermixing and partial fusion of the two bilayers, startingfrom the first interbilayer flip until the nucleation of the hemifused patch. Asabove, the duration of process β defines the reordering time tβ, and (iii) sub-process γ corresponds to the rupture of the hemifused patch, which leads tothe opening of the fusion pore and defines the rupture time tγ. By definition,the total fusion time is given by the sum tfu = tα + tβ + tγ. The averageduration of the sub-processes α and β, 〈tα〉 and 〈tβ〉, are displayed togetherwith 〈tfu〉a as a function of the molecular area in Figure 11. Clearly, both timescales decay exponentially with increasing A and thus with increasing tension.The duration of sub-process γ, on the other hand, appears to be relativelyindependent of both tension and vesicle size. The hemifused diaphragms inall fusion events that do not involve a (meta)stable hemifused state rupturewithin a time interval of 150 to 300 ns.

An improved fit of the average fusion time 〈tfu〉a thus consists of twosuperimposed exponentials plus a constant for the rupture time, as also shownin Figure 11. While at high tensions it agrees with an exponential fit to thefusion time, at low tensions it deviates towards longer times and coincideswith 〈tβ〉.

The time dependence of the flipping and the reordering process impliesthat the corresponding energy barriers governing the flipping rate and thenucleation of the hemifused diaphragm should depend linearly on the mem-brane tension as ∆Eα = ∆Eα,0 − AαΣ and ∆Eβ = ∆Eβ,0 − AβΣ. Here∆Eα,0 and ∆Eβ,0 are the respective barriers for a tension free membraneand Aα and Aβ are characteristic areas. The tension dependence of thecorresponding time scales is described by 〈tα〉 = tsc exp

[

∆Eα,0 − AαΣ]

and〈tβ〉 = tsc exp

[

∆Eβ,0 − AβΣ]

for process α and β, respectively. The expo-nential fits to the simulation data lead to values of the characteristic areasAα ≃ 0.19 and Aβ ≃ 0.69 for the 14 nm and Aα ≃ 0.17 and Aβ ≃ 1.12 for the28 nm vesicle. Numerical values for the energy barriers ∆Eα,0 and ∆Eβ,0 willbe discussed in the next subsections. The dependence of the flipping time onthe tension is much weaker than that of the reordering time. Consequently,as tension is lowered, the fusion time is more and more dominated by thereordering time.

These simulations show that the fusion process has at least three sub-processes with corresponding energy barriers, two of which depend on themembrane tension. To find the magnitude of these energy barriers for relaxedmembranes, the exponential fits of Figure 11 can be extrapolated to zerotension. However, to find the values of ∆Eα and ∆Eβ, knowledge of the


pre-exponential scaling factor, tsc is also required.

The Flipping Sub-Process

Sub-process α is the process of interbilayer flips of single lipid tails. Thislocal process, which changes the conformation of a single lipid relative toits surroundings, is accessible to direct simulation. The energy barrier ∆Eα

for this process is provided by the (partially) hydrated polar head groupsof the proximal monolayers. It is intuitively clear that this barrier shoulddecrease with increasing tension Σ, as the tension causes the head groups inthe planar membrane to move further apart and thus makes it easier for thehydrophobic chains to cross from one bilayer to the other.

To measure this energy barrier, simulations enforcing such interbilayerflips have been performed. In two adhering membranes, a single lipid tailis pulled slowly from its original position into the other bilayer, so that thelipid has one tail in each bilayer as observed in the fusion simulations. Theaverage work required for this process in 20 independent simulations wasfound to be 〈W 〉 = 9 ± 2 kBT . This value constitutes an upper bound forthe energy barrier ∆Eα,0 and should correspond to the barrier itself for veryslow pulling.

Another estimate for the energy barrier comes from the Jarzynski rela-tion (54) as given by

exp

[

−∆F

kBT

]

=⟨

exp

[

−W

kBT

]

⟩

. (5)

This equality should hold irrespective of how fast the process happens, ifa sufficiently large number of trajectories is sampled. The average value

〈exp[

−Wα

kBT

]

〉 from the enforced lipid flip simulations gives a barrier height of

8 kBT .The distribution of the expended work W is relatively wide (See Supple-

mentary Material, Fig. S7), because both the bead in the potential and thebarrier itself fluctuate. Therefore, it is hard to draw any definite conclusionabout the shape of the work distribution from the data.

In order to check the consistency of our previous estimate for the av-erage expended work, we now use a cumulant expansion of the quantity〈exp [W/kBT ]〉. Truncating this expansion at second order, we obtain the


average work

〈W 〉/kBT ≈ ln [〈exp [−W/kBT ]〉]

+1

2〈(W/kBT − 〈W/kBT 〉)

2〉 (6)

which leads to 〈W 〉/kBT ≈ 8.9. This estimate is rather close to the previouslymentioned value 〈W 〉/kBT = 9±2 as obtained by directly averaging the work,which shows that the higher order contributions of the cumulant expansionare small. For the very slow pulling speeds as used here, one would expectan essentially Gaussian distribution, for which the cumulants Cn = 0 forn > 2, and the results of the cumulant expansion agree very well with thisexpectation.

Finally, the decrease of the flipping times 〈tα〉 with increasing tensionas observed in the fusion simulations indicates that the energy barrier forthe flipping process is tension dependent. Additional simulations of enforcedinterbilayer flips, in which the area per molecule A is varied systematically,reproduce this tension dependence, as shown in Figure 12 for two differentvalues of the force amplitude aHC.

Energy Barrier for Interbilayer Flips

The energy barrier for the flipping process, which is due to hydration inreal membranes, is implemented in the coarse-grained simulations in formof the stronger force amplitude aHC between head (H) and tail (C) beads.Therefore, for a given lipid architecture and parameter set, the height of thebarrier can be expected to be determined by the value of the aHC.

A series of simulations of enforced interbilayer flips with different valuesof the aHC parameter in the range aHC = 35 to aHC = 50 confirms thatexpectation. Figure 13 shows both the expectation value 〈W 〉 (blue) andthe energy barriers resulting from the Jarzynski relation (red) determinedin these simulations as a function of aHC. The bars on the Jarzynski pointsindicate the addition of the second cumulant and show that the higher ordersare small for all points. The plots clearly show that the height of the energybarrier increases with increasing strength of aHC.

Since the flipping barrier depends on aHC, its magnitude can be tuned insuch a way that the energy barrier is consistent with available reference data.A possible experimental estimate of the barrier height can be deduced fromthe hydration energy of the hydrocarbon chains, which can be estimated from


the critical micelle concentration (CMC). The hydration energy of DMPC is∼ 23.4 kBT , thus the hydration energy for only one of the hydrocarbon tails,roughly half of this estimate, should be of the order of 10 kBT . Pulling ontwo tails simultaneously in the simulations confirms that this costs abouttwice as much energy as to flip a single one.

Potential of mean force calculations in atomistic MD (55) and experi-mental studies (56, 57) of the partitioning of hexane, i.e. parts of lipid tailswith six carbon atoms, between water and the hydrophobic interior of DOPCbilayers also estimate the transfer energy to be of the order of 10 kBT . Inmany fusion experiments the membranes are strongly dehydrated, see e.g.(58, 59) and this barrier for interbilayer flips might be even lower.

Fusion simulations using the old DPD parameter set (8), see table 1(a)gave different results both for the pathway and statistics: (i) Fusion startsnot at the contact line, but somewhere within the contact area; (ii) Successfulfusion events occur very fast, usually within 300 ns and no tension depen-dence of the fusion time distributions could be deduced; (iii) The successrates are comparatively low, with hemifusion as the most likely result over alarge range of tensions.

Simulations of enforced interbilayer flips show that there is no appreciablebarrier for inter-bilayer flips for the old DPD parameter set (data not shown).The adhered state is unstable and membranes start to intermix upon contact.Furthermore, the membranes are very stable against pore formation and canbe stretched by 60% before rupture.

Comparison of the two parameter sets shows that the aHC parameter ofthe old set is much higher than that of the new set used here, which leads tothe conclusion, that it is not the aHC parameter alone, but rather the ratioaHC/aCC that determines the flipping barrier. For the old and new parameterset, this ratio is aHC/aCC = 2 and aHC/aCC = 3.5, respectively. In the absenceof this barrier, lipids can intermix freely, so that the kink at the contact linewhich serves to lower the flipping energy has no influence on the initiationof fusion. Hemifusion can form rapidly and relax the tension. Fusion thenonly succeeds for very high tensions or when it is rapid enough to proceedbefore hemifusion has completely relaxed the membrane tension. Combinedwith the great stability against pore formation, this serves to stabilize thehemifused state.

If the aHC parameter of the old parameter set is raised by only 5, i.e.aHC = 55 with all other parameters unchanged as in Table 1(b), the fusionpathway no longer proceeds without a barrier. Fusion starts at the contact


line as observed in the simulations here and the fusion time increases from250 to 600 ns. To determine the tension dependence, more simulations atdifferent tensions would be required.

The parameter set used by Gao et al (27) to study the fusion of twovesicles represents much stiffer bilayers where the tail chains tend to align andstick together. This makes splayed lipid conformations much more unlikelyand thus leads to different fusion mechanisms in which the tail chains remainmore aligned.

Additional Energy Barriers

Since the energy barrier for one of the processes of fusion could be measuredin independent simulations, the scale factor tsc has become accessible. Fromthe waiting time 〈tα〉 at Σ = 0 and the estimate of ∆Eα,0 ≃ 8 kBT obtainedfrom the enforced interbilayer flips, the prefactor is found to be tsc = 0.12 nsand tsc = 0.16 ns for the 28 nm and 14 nm vesicle, respectively.

Using these values, we can now estimate the energy barriers for sub-process β corresponding to the nucleation of the hemifused patch, and forsub-process γ describing the rupture of this patch. The energy barrier forsub-process β is found to be ∆Eβ,0 = 11.1±2 kBT and ∆Eβ,0 = 14.4±2 kBTfor the 14 nm and 28 nm vesicle respectively, the barrier for sub-process γ isestimated as 8 kBT .

At low tensions the total fusion time tfu is dominated by the reorderingtime, 〈tβ〉 ,see Figure 11, so that the energy barrier for fusion at low mem-brane tensions will also be dominated by the barrier ∆Eβ for the reorderingprocess. Thus the simulation statistics presented here suggest that the mainenergy barrier for fusion of tensionless bilayers is size dependent and of theorder of 11 kBT and 14 kBT for fusion of the 14 nm and 28 nm vesiclesrespectively.

Dependence of Fusion Times on Vesicle Radius

Similar as for the success rates, there is a difference between the statisticsfor the two vesicle sizes. A comparison of the two graphics shows that thefusion times for the 30 nm vesicle are much longer than those for the 14 nmvesicle. This size dependence originates from the reordering process. Thecharacteristic area Aβ for the 28 nm vesicle is almost a factor two larger thanthat for the 14 nm vesicle, while the flipping time 〈tα〉 and the rupture time


〈tγ〉 appear to be independent of vesicle size.We now want to argue that the different fusion behavior of the two vesicle

sizes is again related to the ratio Rve/L‖ of the vesicle radius to the size ofthe planar membrane, or more precisely to the area difference, N(A − A0),i.e. the difference between the actual area of the stretched membrane andthat of a relaxed membrane with the same number of lipids.

Similar to the analysis by Tolpekina (34), one would have to add N(A−A0)/A lipid molecules in order to relax the tension of the planar membrane.In our system, these lipids can be added via interbilayer flips from the vesicle,and the subsequent formation of a hemifused patch.

If the area difference N(A − A0) is large compared to the vesicle area,the number of lipids required to reduce the tension in the planar bilayerrepresents a considerable part of the vesicle membrane and will perturb thevesicle strongly. For a larger vesicle, this number represents a much smallerfraction of the vesicle’s lipids and thus requires a much smaller perturbationof the membrane. This dependence indicates, that the fusion probability andtime presumably depend on the ratio Rve/L‖.

In order to explicitly confirm this conclusion, additional simulations ofa small vesicle with diameter 14 nm in a smaller simulation box with baselength L‖ = 36 nm have been performed. For this system, the ratio Rve/L‖

is the same as for the system with the 28 nm vesicle. The correspondingdata for the two systems are displayed in Figure S8 of the SupplementaryMaterial. By comparing these data with those of Figure 11, we concludethat the different fusion times observed for the two vesicle sizes are primarilydetermined by the ratio Rve/L‖ rather than the vesicle size Rve alone. Thedifference that can be seen in Figure S 8 between the data for (L‖, Rve) =(36, 20) and (L‖, Rve) = (72, 40) is likely to stem from finite size effects.

Tension-dependence of fusion probability

In the previous subsections, we have focused on the successful fusion eventsand analyzed the corresponding fusion times. These times decrease exponen-tially with increasing membrane tension, see Fig. 10 and 11. Extrapolationto small tensions then leads to the quoted energy barriers for tensionlessmembranes. It is important to note, however, that these barriers apply onlyto the fusion pathway and do not take any other pathway into account as ob-served during the unsuccessful fusion attempts. As mentioned before, thesealternative pathways consist primarily of rupture at large membrane tensions


and of adhesion and hemifusion at small tensions.The fraction of successful fusion events, which defines the fusion proba-

bility, is shown in Fig. 14 as a function of the molecular area A, which isdirectly related to the tension Σ. Inspection of this figure shows that thefusion probability exhibits a maximum at intermediate tension values andthen has a value close to one. For larger tensions, the fusion probabilitydecreases because of rupture; for smaller tensions, it also decreases becauseof adhesion and hemifusion. Linear extrapolation of the simulation data tosmaller values of the molecular area A suggests that the fusion probabilitymay vanish before the membrane reaches its tensionless state. Indeed, nospontaneous fusion events of the smaller vesicle are observed for runs up to20 µs, i.e. twice as long as predicted from the extrapolation in Figure 11.

Thus one may speculate that the system exhibits a threshold value, (Ath−A0), for the membrane stretch, which also implies a threshold tension Σth:for A < Ath, or Σ < Σth, the unfused state is the globally stable one. Sucha threshold is not implausible: If a relaxed bilayer takes up lipids from thevesicle or forms a hemifused patch, it does not lower its energy, but ratherhas to be compressed.

Conclusions

From a detailed study of the effects of individual simulation parameters onthe properties of simulated bilayers, it was possible to construct a coarsegrained membrane with more realistic properties. In particular, the improvedstretching behavior and the introduction of an energy barrier against inter-bilayer flips between bilayers have led to a different fusion pathway, and arealistic dependence of the fusion time on the membrane tension.

A large number of fusion simulations show a common fusion pathway,which involves much more disordered and less symmetric intermediate statesthan is typically assumed. In this process, lipids in a splayed tail conforma-tion, with one tail inserted in each membrane play an important role duringthe onset of fusion.

The timescales of the fusion events suggest that the fusion process consistsof at least three consecutive sub-processes: Interbilayer flips of lipid tails, nu-cleation of a small hemifused area, and pore formation. The first two of thetimescales of the simulated membranes depend exponentially on the tensionsuggesting two tension dependent energy barriers. Using simulations that


enforce interbilayer flips of individual lipid tails and utilizing two differentmethods, one of which makes use of the Jarzynski relation, the energy scalefor these barriers could be determined. Furthermore, one simulation param-eter that is closely related to the energy scale of the interbilayer flips couldbe identified and can be used to tune this energy barrier to a desired size.

These simulations reveal that the conformations of individual moleculesplay a crucial role in membrane processes on such small scales and cannoteasily be neglected. Furthermore the study demonstrates that the fusionprocess has a stochastic character reflecting the thermal fluctuations of thelipid and water molecules. Therefore, in order to obtain reliable data on theaverage fusion times and other observable quantities, it is necessary to studya large number of fusion events. Indeed, both the average fusion time andthe width of the fusion time distribution grow exponentially as one lowers themembrane tension. Because of the large width of the distributions, a singletrajectory does not suffice to reveal the dependence of the average behavior.

For further studies of this kind it would be highly desirable to obtainexperimental data on intermembrane flips and the corresponding energy bar-rier. It could then be possible to extend these simulations to more complexsystems containing mixtures of lipids with realistic properties and includingpossible effects of proteins or other fusogens.

Acknowledgments MEMPHYS is supported by the Danish NationalResearch Council.

References

1. Tamm, L. K., J. Crane, and V. Kiessling, 2003. Membrane fusion: astructural perspective on the interplay of lipids and proteins. CurrentOpinion In Struct. Biol. 13:453–466.

2. Lindau, M., and G. A. de Toledo, 2003. The fusion pore. Biochimica EtBiophysica Acta-molecular Cell Research 1641:167–173.

3. Noguchi, H., and M. Takasu, 2001. Fusion pathways of vesicles: A Brow-nian dynamics simulation. J. Chem. Phys. 115:9547–9551.

4. Muller, M., K. Katsov, and M. Schick, 2002. New mechanism of mem-brane fusion. J. Chem. Phys. 116:2342–2345.


5. Marrink, S., and A. Mark, 2003. Mechanism of vesicle fusion as revealedby molecular dynamics simulations. J. Am. Chem. Soc. 125:11144–11145.

6. Stevens, M. J., J. H. Hoh, and T. B. Woolf, 2003. Insights into themolecular mechanism of membrane fusion from simulation: Evidence forthe association of splayed tails. Phys. Rev. Lett. 91:188102.

7. Smeijers, A. F., A. J. Markvoort, K. Pieterse, and A. J. Hilbers, 2006.A detailed look at vesicle fusion. J. Phys. Chem. B 110:13212 – 13219.

8. Shillcock, J., and R. Lipowsky, 2005. Tension-induced fusion of bilayermembranes and vesicles. Nature Materials 4:225–228.

9. Li, D. W., and X. Y. Liu, 2005. Examination of membrane fusion bydissipative particle dynamics simulation and comparison with continuumelastic models. J. Chem. Phys. 122:1749091–1749098.

10. Knecht, V., A. E. Mark, and S. J. Marrink, 2006. Phase behaviorof a phospholipid/fatty acid/water mixture studied in atomic detail.J.AM.CHEM.SOC. 128:2030–2034.

11. Knecht, V., A. E. Mark, and S. J. Marrink, 2007. Molecular Dynam-ics Simulations of Lipid Vesicle Fusion in Atomic Detail. Biophys. J.92:4254–4261.

12. Kasson, P. M., N. W. Kelley, N. Singhal, M. Vrljic, A. T. Brunger, andV. S. Pande, 2006. Ensemble molecular dynamics yields submillisecondkinetics and intermediates of membrane fusion. PNAS 103:11916–11921.

13. Kasson, P. M., A. Zornorodian, S. Park, N. Singhal, L. J. Guibas, andV. S. Pande, 2007. Persistent voids: a new structural metric for mem-brane fusion. Bioinformatics 23:1753–1759.

14. Grafmuller, A., J. Shillcock, and R. Lipowsky, 2007. Pathway of mem-brane fusion with two tension-dependent Energy barriers. Phys. Rev.Lett 98:218101.

15. Goetz, R., and R. Lipowsky, 1998. Computer simulations of bi-layer membranes:Self-assembly and interfacial tension. J. Chem. Phys.108:7397–7409.


16. Shillcock, J., and R. Lipowsky, 2002. Equilibrium structure and lat-eral stress distribution of amphiphilic bilayers from dissipative particledynamics simulations. J. Chem. Phys. 117:5048–5061.

17. A. Finkelstein, J. Z., and F. S. Cohen, 1986. Osmotic swelling of vesicles:its role in the fusion of vesicles with planar phospholipid bilayer mem-branes and its possible role in exocytosis. Anu. Rev. Physiol. 48:163–174.

18. Rowlinson, J. S., and B. Widom, 1982. Molecular Theory of Capillarity.Oxford, Oxford University Press, Walton Street, Oxford OX2 6DP.

19. Groot, R. D., and P. B. Warren, 1997. Dissipative particle dynamics:Bridging the gap between atomistic and mesoscopic simulation. J. Chem.Phys. 107:4423–4435.

20. Vattulainen, I., M. Karttunen, G. Besold, and J. M. Polson, 2002. In-tegration schemes for dissipative particle dynamics simulations: Fromsoftly interacting systems towards hybrid models. J. Chem. Phys.116:3967–3979.

21. Yamamoto, S., Y. Maruyama, and S.-A. Hyodo, 2002. Dissipative par-ticle dynamics study of spontaneous vesicle formation of amphiphilicmolecules. J. Chem. Phys. 116:5842–5849.

22. Kranenburg, M., M. Venturoli, and B. Smit, 2003. Phase Behavior andInduced Interdigitation in Bilayers Studied with Dissipative Particle Dy-namics. J. Phys. Chem. B 107:11491–11501.

23. Stoyanov, S. D., and R. D. Groot, 2004. From molecular dynamics tohydrodynamics: A novel Galilean invariant thermostat. J. Chem. Phys.122:114112/1 – 114112/8.

24. Guillot, B., 2002. A reappraisal of what we have learnt during threedecades of computer simulations on water. JOURNAL OF MOLECU-LAR LIQUIDS 101:219–260.

25. Kornberg, R. D., and W. Mcconnel.hm, 1971. Inside-outside transitionsof phospholipids in vesicle membranes. Biochem. 10:1111.

26. Wimley, W. C., and T. E. Thompson, 1990. Exchange and flip-flop of dimyristoylphosphatidylcholine in liquid-crystalline, gel, and 2-component, 2-phase large unilamellar vesicles. Biochem. 29:1296–1303.


27. Gao, L., R. Lipowsky, and J. Shillcock, 2008. Tension-induced vesiclefusion: pathways and pore dynamics. Soft Matter 4:1208–1214.

28. Schofield, P., and J. R. Henderson, 1982. Statistical Mechanics of Inho-mogeneous Fluids. Proc. R. Soc. London A 379:231–246.

29. Nagle, J. F., and S. Tristam-Nagle, 2000. Structure of lipid bilayers.Biochimica et Biophysica Acta 1469:159–195.

30. Oradd, G., G. Lindblom, and P. W. Westermann, 2002. Latteral Diffu-sion of Cholesterol and Dimyristylphosphatidylcholine in Lipid BilayerMeasured by Pulsed Field Gradient NMR Spectroscopy. Biophys. J.83:2702–2704.

31. Ortiz, V., S. O. Nielsen, D. E. Discher, M. L. Klein, R. Lipowsky, andJ. C. Shillcock, 2005. Dissipative Particle Dynamics Simulations of Poly-mersomes. J. Phys. Chem. B 109:17708–17714.

32. Evans, E., and W. Rawicz, 1990. Entropy Driven Tension and BendingElasticity in Condensed-Fluid Membranes. Phys. Rev. Lett 64:2094–2097.

33. Discher, B. M., Y.-Y. Won, D. S. Ege, J. C.-M. Lee, F. S. Bates, D. E.Discher, and D. A. Hammer, 1999. Polymersoms: Tough Vesicles Madefrom Diblock Copolymers. Science 284:1143–1146.

34. Tolpekina, T. V., W. K. den Otter, and W. J. Briels, 2004. Simulationsof stable pores in membranes: System size dependence and line tension.J. Chem. Phys. 121:8014 – 8020.

35. Illya, G., R. Lipowsky, and J. Shillcock, 2005. Effect of chain lengthand assymmetry on material properties of bilayer membranes. J. Chem.Phys. 122:244901.

36. Rawicz, W., K. C. Olbrich, T. McIntosh, D. Needham, and E. Evans,2000. Effect of Chain Length and Unsaturation on Elasticity of LipidBilayers. Biophys. J. 79:328–339.

37. Cantor, R. S., 1997. Lateral pressures in cell membranes: A mechanismfor modulation of protein function. J. Phys. Chem. B 101:1723–1725.


38. Asakura, S., and F. Oosawa, 1958. Interaction of particles suspended insolutions of macromolecules. J. Polymer Sci. 33:183–192.

39. Lis, L. J., M. McAlister, N. Fuller, R. P. Rand, and V. A. Parsegian, 1982.Interactions between neutral phospholipid bilayer membranes. Biophys.J. 37:657–666.

40. Cevc, G., W. Fenzl, and S. L., 1990. Surface-Induced X-Ray ReflectionVisualization of Membrane Orientation and Fusion into Multibilayers.Science 249:1161–1163.

41. Marra, J., and J. Israelachvili, 1985. Direct measurements of forcesbetween phosphatidylcholine and phosphatidylethanolamine bilayers inaqueous-electrolyte solutions. Biochemistry 24:4608–4618.

42. Evans, E., and D. Needham, 1987. Physical Properties of SurfactantBilayer Membranes: Thermal Transitions, Elasticity, Rigidity, Cohesion,and Colloidal Interactions. J. Phys. Chem. 91:4219–4228.

43. Helfrich, W., 1995. Tension-Induced Mutual Adhesion and a ConjecturedSuperstructure of Lipid Membranes. In R. Lipowsky, and E. Sackmann,editors, Handbook of Biological Physics, Elsevier, Amsterdam, p. 691–721.

44. Lipowsky, R., 1995. Generic Interactions of Flexible Membranes. InR. Lipowsky, and E. Sackmann, editors, Handbook of Biological Physics,Elsevier, Amsterdam, p. 521–602.

45. Lipowsky, R., 1986. Melting at Grain Boundaries and Surfaces. Phys.Rev. Lett. 57:2876.

46. Lipowsky, R., and S. Leibler, 1986. Unbinding transitions of interactingmembranes. Phys. Rev. Lett. 56:2541–2544.

47. Seifert, U., and R. Lipowsky, 1990. Adhesion of vesicles . Phys. Rev. A42:4768 – 4771.

48. Kinnunen, P. K. J., and J. M. Holopainen, 2000. Mechanisms of initiationof membrane fusion: Role of lipids. Bioscience Reports 20:465–482.


49. Wang, L., E. S. Seeley, W. Wickner, and A. J. Merz, 2002. Vacuole fusionat a ring of vertex docking sites leaves membrane fragments within theorganelle. Cell 108:357–369.

50. Chernomordik, L., M. M. Kozlov, and J. Zimmerberg, 1995. Lipids inbiological membrane-fusion. J. Membrane Biol. 146:1–14.

51. Chernomordik, L., 1996. Non-bilayer lipids and biological fusion inter-mediates. Chem. Phys. Lipids 81:203–213.

52. Evans, K. O., and B. R. Lentz, 2002. Kinetics of lipid rearrangementsduring poly(ethylene glycol)-mediated fusion of highly curved unilamellarvesicles. Biochem. 41:1241–1249.

53. Lentz, B. R., W. Talbot, J. K. Lee, and L. X. Zheng, 1997. Transbi-layer lipid redistribution accompanies poly(ethylene glycol) treatment ofmodel membranes but is not induced by fusion. Biochem. 36:2076–2083.

54. Jarzynski, C., 1997. Nonequilibrium Equality for Free Energy Differ-ences. Phys. Rev. Lett. 78:2690 – 2693.

55. MacCallum, J. L., and D. P. Tieleman, 2006. Computer simulation of thedistribution of hexane in a lipid bilayer: Spatially resolved free energy,entropy, and enthalpy profiles. J. Am. Chem. Soc. 128:125–130.

56. White, S. H., G. I. King, and J. E. Cain, 1981. Location of hexane inlipid bilayers determined by neutron-diffraction. Nature 290:161–163.

57. Deyoung, L. R., and K. A. Dill, 1990. Partitioning of nonpolar solutesinto bilayers and amorphous n-alkanes. J. Phys. Chem. 94:801–809.

58. Lentz, B. R., 2007. PEG as a tool to gain insight into membrane fusion.European Biophys. J. With Biophys. Lett. 36:315–326.

59. Tamm, L. K., X. Han, Y. Li, and A. L. Lai, 2002. Structure and Functionof Membrane Fusion Peptides. Biopolymers (Peptide Science) 66:249–260.


Figure Legends

Figure 1.

A coarse-grained model dimyristoyl-phosphatidylcholine (DMPC) with a H3(C4)2

architecture consisting of 3 head (H) beads and two hydrocarbon chains eachconsisting of chain (C) beads. Each chain bead C represents 3.5 CH2 groups.Consecutive beads are connected by springs of unstretched length l0. Thehydrophobic chains are stiffened by a three body potential constraining theangle ψ between two consecutive bonds.


Figure 2.

Bilayer tension Σ as a function of area per molecule A calculated from thestress profile s(z) for the old DPD parameter set (8, 16) (circles) and thenew parameter set with improved stretching behavior (diamonds). For thenew parameter set, Σ has also been measured by direct z-integration of themicroscopic stress tensor (crosses). Error bars represent the standard error.The two methods are found to yield the same results. The two parametersets are given in Table 1.


Figure 3.

Simulations enforcing interbilayer flips of lipid tails are used to measure theenergy barrier ∆Eα for interbilayer flips. (a) From two adhering bilayers(head beads blue/green, tail beads omitted for clarity) a single lipid is selected(orange heads, yellow/red tails) and a slowly moving external harmonic forceF applied to one of its tail beads (red), until the tail has flipped to the otherbilayer, so that the lipid has assumed a splayed configuration with one tailinserted in each bilayer as shown in (c). (b) The energy landscape Eα for thebead is sketched as a function of the displacement z of the yellow bead. Ithas a high barrier in the center corresponding to the repulsive head groupsand increases to the sides reflecting displacement of the head group into thehydrophobic region.


Figure 4.

Membrane tension Σ as a function of molecular area A for three differentchain lengths of 3,4 and 5 per tail chain, in the old (a) and new (b) parameterset. There is no state with zero tension for bilayers formed from lipids with3 beads per chain in the old parameter set (a), since at low A values thebilayer structure is not stable.


Figure 5.

Fusion of a vesicle with a diameter of 28 nm to a planar membrane with aprojected area of (50nm)2. The vesicle consists of 6869 lipids (orange heads,yellow chains) while the planar membrane contains 6911 lipids (red heads,green chains). The water beads originally inside the vesicle are blue, thoseoutside are not shown for clarity. Six snapshots illustrating the developmentof the fusion event from 78.5 ns after the first contact until opening of thefusion pore after 1334 ns. For each time, the system is shown from two per-spectives: cross sections cut through the center of the vesicle viewed from theside, and two cross sections through the midplane of the planar membrane,as indicated by the arrows in (a) viewed from above. In the upper top views,the green hydrophobic beads from the planar bilayer are made transparent,revealing the yellow hydrophobic chains of vesicle lipids which have flippedinto the planar bilayer. In the lower top views, all hydrophobic beads areset transparent, so that white areas in the head group plane indicate purelyhydrophobic areas. Lipid tails start to undergo interbilayer flips after 78.5ns. The growth of the contact area enhances these at the contact line, in-dicated by the blue broken line in (b), creating a bean-shaped, disorderedhydrophobic contact which nucleates into a hemifused diaphragm after 1177ns.


Figure 6.

(a) Example of a vesicle that adheres to a planar bilayer patch. Both thevesicle and the deformed segment of the planar membrane are well fitted byspherical caps. (b) The dependence of the adhesion energy density W on thedifference (A− A0) of the area per molecule from its tensionless value. Eachdata point represents the average over data from 10 to 20 different snapshots.Error bars are one standard deviation and the solid line is the best linear fit.


Figure 7.

The first interbilayer flips of the vesicle’s hydrophobic chains (yellow) into theplanar bilayer (head beads red, chain beads not shown) (a) Top view of theplanar membrane, constructed the same as the upper topviews in Figure 5.The hydrophobic chains and the upper monolayer of the planar bilayer aretransparent, revealing the yellow hydrophobic beads of vesicle lipids thathave flipped into the planar membrane. The rate of interbilayer flips is low,so that the influence of the contact line on the probability for interbilayer flipsbecomes clearly visible. The arrow indicates a flipped lipid that has diffusedaway from the contact area. At slow flipping rates this diffusion competeswith the flipping. (b) Snapshots of the center of the planar membrane, withits hydrophobic chains set transparent. These snapshots show the detailsof the first hydrophobic chains belonging to vesicle lipids (yellow) movinginto the planar bilayer. At first only one tail flips, so that the lipid assumesa splayed conformation. Further lipids undergo the same transition in thevicinity, presumably because the splayed lipid sufficiently disturbs the bilayerstructure. In the final snapshot, one lipid has flipped both of its tails intothe other bilayer.


Figure 8.

(a) An example of the disordered domain at the contact line, side view. Thehydrophobic material of the two bilayers (green/yellow) is no longer separatedby a head group layer (red/orange) and has mixed. Several orange headbeadsare trapped in the center of this region. (b) Top view onto the planar bilayerpatch. A small region of head beads from vesicle lipids (orange) appearsbetween the planar bilayer lipids (red), indicating a region where severallipids from the vesicle have moved across the planar bilayer.


Figure 9.

A 14 nm vesicle and part of a 50 nm planar bilayer at A = 1.45, whichhave formed an extended hemifused contact. Only the central part of thesimulation box is shown. A small pore, indicated by the arrow, has formedat the junction of the three bilayers. Such pores allow the pressure of theenclosed water to be reduced and fast lipid flip-flops between the inner andouter monolayers to occur.


Figure 10.

The average fusion times 〈tfu〉a (filled diamonds) and 〈tfu〉b (open diamonds)as functions of the area per molecule A displayed together with the widths∆tfu of the fusion time distributions (crosses) (a) for the 14 nm and (b) forthe 28 nm vesicle. The two averages 〈tfu〉a and 〈tfu〉b represent an upper andlower bound for the average fusion time 〈tfu〉. Both 〈tfu〉a and ∆tfu seem todecrease exponentially with A.


Figure 11.

The average duration of the tension dependent sub-processes 〈tα〉 (red circles)and 〈tβ〉 (green diamonds) displayed together with 〈tfu〉a (blue diamonds) asa function of the area per molecule A (a) for the 14 nm and (b) for the 28nm vesicle. Both 〈tα〉 and 〈tβ〉 show an exponential dependence on A. Thelight blue curve represents a new fit of the fusion time based on the sum〈tα〉+ 〈tβ〉+ 〈tγ〉, where 〈tγ〉 is the rupture time of the hemifused diaphragm.


Figure 12.

The energy barrier ∆Eα for the interbilayer flip of a lipid tail as a functionof the area per molecule A for two values of the head tail force amplitudeaHC, aHC = 50 (blue diamonds) and aHC = 35 (red circles). Each point isthe average of 20 independent enforced interbilayer flips and the error barsrepresent one standard deviation. In the fusion simulations, the parametervalue aHC = 35 has been used. The area A0 = 1.25 corresponds to thetensionless membrane.


Figure 13.

Diamonds: The average energy barrier height plotted against the strengthof the head-tail force amplitude aHC. Each point is determined from 20independent enforced interbilayer flips. The error bars represent one standarddeviation. Circles: the energy determined by the Jarzynski relation 5. Theend of the upward bar indicates the average expended work up to second orderof the cumulant expansion. The average barrier height clearly increases withlarger values of aHC, indicating that the strength of head tail repulsion of alipid can ‘tune’ the flipping energy.


Figure 14.

Fusion probability as a function of molecular area A for (a) the 14 nm and (b)the 28 nm vesicles. In both cases, the fusion probability, which representsthe fraction of fusion attempts that lead to fusion within 20 µs, exhibitsa maximum at Amax with 1.45 < Amax < 1.5 in (a) and Amax ≃ 1.5 in (b)corresponding to the tensions Σ ≃ 3.36 and Σ ≃ 4.25, respectively. At highertensions, fusion becomes less likely because of membrane rupture; at lowertensions, fusion is more and more replaced by adhesion or hemifusion. Alinear extrapolation of the data to smaller values of A indicates a moleculararea threshold for fusion at Ath = 1.29 for the 14 nm and Ath = 1.36 for the28 nm vesicle. This corresponds to a tension threshold Σth ≃ 0.56 for the 14nm vesicle and Σth ≃ 1.79 for the 28 nm vesicle.


Figure 1.

Figure 1:


Figure 2.

Figure 2:


Figure 3.

Figure 3:


Figure 4.

Figure 4:


Figure 5.

Figure 5:


Figure 6.

Figure 6:


Figure 7.

Figure 7:


Figure 8.

Figure 8:


Figure 9.

Figure 9:


Figure 10.

Figure 10:


Figure 11.

Figure 11:


Figure 12.

Figure 12:


Figure 13.

Figure 13:


Figure 14.

Figure 14:

The fusion of Membranes and Vesicles: Pathway and Energy ... · The Fusion of Membranes and...

Documents

Transcript of The fusion of Membranes and Vesicles: Pathway and Energy ... · The Fusion of Membranes and...