Coarse-grained molecular dynamics simulationof binary charged lipid membranes:

Phase separation and morphological dynamics

Hiroaki Ito∗

Department of Mechanical Engineering, Graduate School of Engineering, Osaka University, Osaka 565-0871, Japan

Yuji HiguchiInstitute for Materials Research, Tohoku University, Miyagi 980-8577, Japan

Naofumi ShimokawaSchool of Materials Science, Japan Advanced Institute of Science and Technology, Ishikawa 923-1292, Japan

(Dated: September 5, 2018)

Biomembranes, which are mainly composed of neutral and charged lipids, exhibit a large varietyof functional structures and dynamics. Here, we report a coarse-grained molecular dynamics (MD)simulation of the phase separation and morphological dynamics in charged lipid bilayer vesicles. Thescreened long-range electrostatic repulsion among charged head groups delays or inhibits the lateralphase separation in charged vesicles compared with neutral vesicles, suggesting the transition of thephase-separation mechanism from spinodal decomposition to nucleation or homogeneous dispersion.Moreover, the electrostatic repulsion causes morphological changes, such as pore formation, andfurther transformations into disk, string, and bicelle structures, which are spatiotemporally coupledto the lateral segregation of charged lipids. Based on our coarse-grained MD simulation, we proposea plausible mechanism of pore formation at the molecular level. The pore formation in a charged-lipid-rich domain is initiated by the prior disturbance of the local molecular orientation in thedomain.

I. INTRODUCTION

Biomembranes, which are formed by self-assembly ofvarious amphiphilic molecules, exhibit a large variety offunctional structures and dynamics because these mem-branes are small, soft, and heterogeneous. There isdynamic compositional heterogeneity in biomembranes,namely, the transient formation of small domains en-riched in saturated lipids and cholesterol, known as lipidrafts [1, 2]. Raft domains may play important rolesin signal transduction and membrane trafficking [3–5].To study the physicochemical mechanism of this lat-eral membrane heterogeneity, phase separation in multi-component lipid membranes, especially in giant unilamel-lar vesicles (GUVs) consisting of mixtures of saturatedand unsaturated lipids, has attracted a great deal of at-tention as a model in vitro.

Typically, in ternary lipid mixtures composed of satu-rated lipids, unsaturated lipids, and cholesterol, liquid-ordered (Lo) and liquid-disordered (Ld) phases coex-ist [6–8]. In particular, the equilibrium conditions forstable phase separation [9, 10] and the domain-growthdynamics after the sudden transition into coexistenceconditions [11–13] have been extensively investigated,confirming both the equilibrium and dynamic featuresof phase separation in electrically neutral systems. Re-cently, there has been increasing interest in the effects oflong-range electrostatic interactions on the phase separa-

∗ ito@hh.mech.eng.osaka-u.ac.jp

tion [14–18] because plasma membranes [19, 20] and someorganelles, such as lysosomes [21] and mitochondria [22],are enriched with charged lipids. In experiments withsystems containing charged lipids, charged unsaturatedlipids tend to suppress the phase separation comparedwith neutral lipid systems [14–18], whereas charged satu-rated lipids promote phase separation [18]. Several theo-retical models [23–26] have also been proposed to explainthe electrostatic effects on the lateral phase separation incharged lipid membranes.

Morphological changes in lipid bilayer membranes areimportant dynamic phenomena in biomembranes. Ex-amples include the budding in endo- and exocytosis [27],the fission-fusion sequence of vesicular transport [28],and pore formation in autophagy [29]. Inspired bythe dynamic changes in biomembrane morphology, stud-ies have also investigated the morphological changes inGUVs induced by external stimuli, such as osmotic pres-sure [30], addition of surfactant molecules [31], and asynthetic photosensitive amphiphile [32]. Some ions orcharged biomolecules can also trigger dynamic morpho-logical changes through adsorption onto the lipid mono-layers or bilayers via the electrostatic interactions. Forexample, locally injected HCl drives cristae-like defor-mation [33], BAR (Bin/amphiphysin/Rvs) domain fam-ily proteins induce the formation of tubular structures onthe membranes [34], and actin cytoskeletons with myosinmotor proteins cause concave deformations of oppositelycharged lipid membranes [35, 36].

In biomembranes, phase separation and morphologicalchanges in the membrane are closely coupled. Recently,raft domains have been reported to play a critical role in

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membrane dynamics, such as autophagy and endocyto-sis, in response to chemical signals [37, 38]. Using GUVscomposed of neutral lipids, coupling between phase sep-aration and morphological dynamics has also been ob-served. For example, the adhesion between GUVs con-sisting of inverse cone- and cylinder-shaped lipids [39],the pore formation in membranes consisting of cone-and cylinder-shaped lipids [40], and the coupling be-tween the phase separation and deformed membranesby osmotic pressure [41, 42] have been reported. Sev-eral studies have focused on the coupling between thephase separation and the morphology in charged lipidmembranes. For instance, a positively charged protein,cytochrome c, induces the collapse of domains enrichedwith cardiolipin (CL(−)), which is a negatively chargedunsaturated lipid abundant in the inner membrane ofmitochondria [43]. Even for more physiologically rele-vant phospholipids with ions, the domains enriched withthe unsaturated negatively charged lipid (dioleoylphos-phatidylserine; DOPS(−)) bud toward the interior of theGUV upon the addition of calcium ions [15], and thoseenriched with the saturated charged lipids (dioleoylphos-phatidylglycerol; DOPG(−)) undergo pore formation inthe membrane by electrostatic repulsion [44].

Although phase separation and the morphologies ofcharged lipid membranes have been extensively investi-gated, there are still several questions about the mecha-nism of charge-specific dynamics. For example, althoughthe domain-coarsening dynamics are frequently examinedin neutral membranes [11–13], these experiments havenot been performed in charged lipid membranes due tothe difficulty in preparing experimental systems in whichthe charge and ion conditions are well-controlled. It isdifficult to obtain reliable statistics because the estab-lished vesicle-preparation method of electroformation isnot applicable to charged systems [16]. To reveal thephase-separation dynamics, morphological changes, andthe coupling between them in charged lipid membranes,we use a coarse-grained molecular dynamics (MD) sim-ulation [45, 46] of binary charged and neutral lipid mix-tures. To capture such mesoscopic and long-time featuresof vesicle dynamics, highly coarse-grained model is nec-essary [47, 48], as we successfully reproduced the experi-mental morphologies of charged lipid vesicles in our pre-vious work using the present simulation model [44]. Thecoarse-grained MD simulation is suitable for this pur-pose because we can specify various conditions, such aslipid composition and salt concentration, we can easilyinvestigate the effects of pure electrostatic interactionson a membrane composed of a number of dynamicallymoving molecules, and we can observe the underlyingmechanism at an adequate molecular level. We investi-gate the dynamic growth of the charged domain and thecoupling between the phase separation and the morpho-logical changes including topological changes. The de-tails of the coarse-grained MD simulation are introducedin Sec. II. The phase separation and the domain growthare investigated in Secs. III A and III B. The morpho-

logical changes induced by electrostatic interactions arepresented in Sec. III C. The implications of our work andcomparisons with other studies are discussed in Sec. IV.

II. METHODS

We extended the coarse-grained model described byCooke et al. [49] to calculate the behavior of charged lipidmembranes. This highly coarse-grained model has beenused to simulate various mesoscopic behaviors [49, 50],where the physical properties of the same orders ofmagnitude with more detailed models have been ob-tained [51]. A lipid molecule is represented by one hy-drophilic head bead followed by two hydrophobic tailbeads. The excluded-volume interaction between twobeads separated by a distance r is

Vrep(r; b) =

{4v[(

br

)12 − ( br )6 + 14

], r ≤ rc,

0, r > rc,(1)

where rc = 21/6b. v is the unit of energy. We chosebhead,head = bhead,tail = 0.95σ and btail,tail = σ to form

a stable bilayer, where σ = 7.09A is the unit of lengthcorresponding to the cross-sectional diameter of a sin-gle lipid molecule. The potentials for the stretching andbending of bonds between connected beads are expressedas

Vbond(r) =1

2kbond(r − σ)2 (2)

and

Vbend(θ) =1

2kbend(1− cos θ)2, (3)

where kbond = 500v/σ2 and kbend = 60v are the bond-ing strength of connected beads and the bending stiffnessof a lipid molecule, respectively. θ is the angle betweenadjacent bond vectors. The hydrophobic attractive in-teraction among hydrophobic beads is expressed as

Vattr(r) =

−v, r < rc,

−v cos2[π(r−rc)

2wc

], rc ≤ r ≤ rc + wc,

0, r > rc + wc,

(4)

where wc is the cut-off length for the attractive poten-tial. The lipid membrane is in the gel phase when wc

is large, whereas it is in the liquid phase when wc issmall. Such gel and fluid phases observed in the bi-nary lipid bilayer membranes are directly relevant to theliquid-ordered (Lo) and liquid-disordered (Ld) phases, re-spectively, which are experimentally observed in ternarylipid bilayer membranes composed of unsaturated lipids,saturated lipids, and cholesterols, as both lateral phaseseparations in the simulation and experiments are rec-ognized by circular domains. Thus, it is possible to de-termine whether the lipid species is in the gel or liquid

3

phase by controlling wc in a neutral lipid system [49]. Torepresent the charged lipids, we considered the electro-static interaction among charged head groups in additionto the other interactions. The electrostatic repulsion isdescribed as the Debye-Huckel potential

Velec(r) = v`Bz1z2exp(−r/`D)

r, (5)

where `B = σ is the Bjerrum length, z1 and z2 are thevalencies of the interacting charged head groups, and`D = σ

√εkBT/n0e2 is the Debye screening length which

is related to the bulk salt concentration n0 (ε, kB, T ,and e are the dielectric constant of the solution, Boltz-mann constant, absolute temperature, and elementarycharge, respectively). To capture the behaviors causedby the screened electrostatic interaction qualitatively, n0is set to 1 M, 100 mM, or 10 mM, varying by an orderof magnitude around the physiological salt concentra-tion of 100 mM. Because the typical negatively chargedhead groups, such as phosphatidylglycerol (PG), phos-phatidylserine (PS), phosphatidylinositol (PI), and phos-phatidyl acid (PA), have a monovalent charge, we setz1 = z2 = −1. We did not set a cutoff for this screenedelectrostatic interaction.

Each bead position ri obeys the stochastic dynamicsdescribed by the Langevin equation

md2ridt2

= −η dridt

+ fVi + ξi, (6)

where m = 1 and η = 1 are the mass and drag coef-ficients, respectively. The force fVi is calculated fromthe derivatives of the interaction potentials Eqs.(1)–(5).The constant τ = ησ2/v is chosen as the unit for thetimescale, and the time increment is set at dt = 7.5 ×10−3τ . The Brownian force ξi satisfies the fluctuation-dissipation theorem

〈ξi(t)ξj(t′)〉 = 6vηδijδ(t− t′). (7)

In this study, we calculated bilayer vesicles composed oftwo lipid species to represent the ternary lipid vesicleswith gel and liquid domains observed in experiments [49].The calculated lipid composition is a binary mixture ofA-lipids with or without a monovalent electric chargeat the head beads, corresponding to charged or neutrallipids, respectively, and neutral B-lipids. Binary neutralmembranes consisting of neutral A- and B-lipids are usedas the reference condition for the binary charged mem-brane consisting of charged A-lipids and neutral B-lipids.The initial state of the calculation is a spherical lipid bi-layer consisting of 5000 lipid molecules in total, whereA- and B-lipids are mixed homogeneously at a specifiedratio, A:B. The total calculation time is set to t = 1.0–10.0× 7500τ , during which the phase-separation or mor-phological dynamics adequately relaxes.

III. RESULTS

A. Lateral phase separation

(a)

t = 0.3t = 0.2t = 0.1

t = 0.75 t = 1.0t = 0.5t = 0.4

t = 0.01(units of 7500τ)

(b)

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-28

-26

-24

-22

0.20 0.4 0.6

time t (units of 7500τ)

0.8 1.0

Va

ttr

/ lip

id (

un

its o

f k

BT

)

w / σ = 1.6

1.651.71.751.8

AAc

FIG. 1. (a) Typical snapshots of the lateral phase separationof a neutral vesicle consisting of 2500 neutral A-lipids [head,red (dark); tail, green] and 2500 neutral B-lipids [head, yellow(light); tail, cyan]. After rapid relaxation on the order oft ∼ 0.1×7500τ , the binary vesicle reaches a quasi-equilibriumstate. Attractive interaction parameters are set to wAA

c /σ =1.7, wBB

c /σ = 1.7, and wABc /σ = 1.5. (b) Time evolution

of the attractive potential Vattr per lipid molecule during thelateral phase separation for t = 1.0 × 7500τ under variousattractive interactions between A-lipids of wAA

c /σ = 1.6, 1.65,1.7, 1.75, and 1.8. Calculations were performed five times foreach wAA

c /σ value to ensure reproducibility.

First, we investigated the dynamic features of lateralphase separation in neutral binary lipid vesicles as thereference behavior for that of charged lipid vesicles. Be-cause both A- and B-lipids are neutral, we set the va-lencies of their head groups as zA = zB = 0. In otherwords, force fV is obtained from Eqs.(1)–(4) without theelectrostatic potential in Eq.(5). Figure 1(a) shows typ-ical snapshots of lateral phase separation in the binarymixture of neutral A-lipids (red or dark) and B-lipids(yellow or light) with A:B = 2500:2500, where the at-tractive interaction parameters were set to wAA

c /σ = 1.7,wBB

c /σ = 1.7, and wABc /σ = 1.5. Immediately after the

start of the simulation, the vesicle was in the laterallyhomogeneous state (t = 0.01 × 7500τ). The phase sep-aration was gradually induced by the attractive interac-tion being stronger among the same species than thatamong different species on a timescale of t = 0.1×7500τ .Figure 1(b) shows the attractive potential, Vattr, per

4

0.35

0.4

0.45

0.5

0.55

0.6

0.65

1.4 1.5 1.6 1.7 1.8 1.9 2

Ne

ma

tic o

rde

r p

ara

me

ter S

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Me

an

-sq

ua

red

diffu

sio

n le

ng

th / σ

1.4 1.5 1.6 1.7 1.8 1.9 2

w / σAAc

w / σAAc

(a)

(b)

FIG. 2. (a) Mean-squared diffusion length of a lipid in 1000dtplotted versus attractive interaction parameter wAA

c /σ. (b)Nematic order parameter S plotted versus wAA

c /σ. The val-ues are averaged spatially over a vesicle and temporally over0.9–1.0 × 7500τ for each wAA

c /σ in both (a) and (b). Er-ror bars represent the standard deviations. Other interactionparameters are set to wBB

c /σ = 1.7, and wABc /σ = 1.5.

lipid molecule during the phase separation for attrac-tion parameters wAA

c /σ = 1.6, 1.65, 1.7, 1.75, and 1.8,where the bilayer membrane phase ranged from fluid togel [49]. We quantitatively confirmed the transition be-tween liquid and gel phases induced by the attractiveinteraction parameter wAA

c at wAAc /σ = 1.7, based on

the calculations of the lipid diffusivity and nematic orderparameter in the vesicle. Figure 2(a) shows the mean-squared diffusion length of a lipid in 1000dt plotted ver-sus wAA

c /σ. Although a smaller wAAc is accompanied by

a larger error resulting from spike noises caused by fre-quent dropouts of lipids, the diffusion length decreasedaccording to the increase in wAA

c , and reached a plateaufor wAA

c /σ ≥ 1.7. We also calculated the nematic orderparameter by S = 1

2 〈3vb,i · vb,j − 1〉i,j , where vb,i is the

unit bond vector of tail beads of the ith lipid and 〈 〉i,jrepresents the average over all the combinations of lipidpairs [Fig. 2(b)]. The nematic order parameter also showsa clear plateau at wAA

c /σ ≥ 1.7, indicating the most or-dered state with the lowest diffusivity in this parameterrange. These two features in Figs. 2(a) and 2(b) char-

acterize the transitional behavior of lipid bilayer mem-branes between gel and liquid phases in our simulation.Despite the phase transition around wAA

c /σ = 1.7, eventhough the values of Vattr were slightly different accord-ing to wAA

c in Fig. 1(b), the vesicles exhibited the samequalitative feature in their dynamics, namely, a decreasein potential on a timescale of t ∼ 0.1× 7500τ . After thisrapid relaxation into phase-separated states, the vesiclesreached quasi-equilibrium states, identified by the flat-tening of the temporal changes in Vattr. Complete phaseseparation with two macro domains is reached stochas-tically on much longer timescales than the practical cal-culation time in this study because of the much slowerdiffusion of large domains compared with single molecule.Because the other kinds of energies Vrep, Vbond, and Vbenddid not exhibit any clear changes during the phase sep-aration, the change in attractive interaction dominatedthe lateral phase separation in neutral lipid vesicles inboth the gel and liquid states.

Next, we considered a negatively charged vesicle con-sisting of negatively charged A-lipids and neutral B-lipidsto examine the effect of long-range electrostatic repulsionon the lateral phase separation. The valencies of the headgroups of A- and B-lipids were set as zA = −1 and zB =0, respectively. Figure 3(a) shows typical snapshots ofthe lateral phase separation of vesicles composed of 2500charged A-lipids and 2500 neutral B-lipids under differ-ent salt concentrations. All the parameters except forthe valency of the A-lipids were the same as in the calcu-lation for the neutral vesicles (Fig. 1). For charged lipidsat a higher salt concentration of n0 = 1 M the vesicles ex-hibited dynamics similar to those of the neutral vesicles,whereas at a lower salt concentration of n0 = 100 mMthey exhibited clearly slower phase-separation dynamics.The attractive potential Vattr of the charged lipid vesi-cles at n0 = 1 M (Fig. 3(b)) was also quantitatively thesame as that of the neutral lipids (Fig. 1(b)), althoughthe electrostatic repulsive potential, Velec, increased withthe phase-separation process (Fig. 3(c)). Thus, the elec-trostatic interaction among charged lipids with zA = −1was negligible at n0 = 1 M, and the increase in Velecwas two orders of magnitude lower than the decrease inVattr. However, at n0 = 100 mM, the phase-separationdynamics were affected by the less-screened electrostaticrepulsion as the decrease in Vattr became slower and Velecincreased by an order of magnitude compared with thatat n0 = 1 M (Fig. 3(d) and 3(e)).

We compared the static phase-separated states of neu-tral and charged vesicles consisting of 2500 neutral orcharged A-lipids and 2500 neutral B-lipids. We focusedon the wAB

c dependence of the static states because theattractive parameter wAB

c governs the degree of mixing,which provides quantitative information about suscep-tibility of the phase separation in neutral and chargedvesicles. As shown in Fig. 4(a), there was a large dif-ference in the the critical attractive parameter, wAB

c , forthe lateral phase separation. The corresponding changesin the attractive interaction potential, Vattr, shown in

5

100 mM

n0 =

1 M

(a)

(b)

(d) (e)

(c)

t = 0.4 t = 0.5 t = 0.75 t = 1.0t = 0.3t = 0.2t = 0.1t = 0.01 (units of 7500τ)

n0 = 1 M n0 = 1 M

n0 = 100 mM n0 = 100 mM

0.20 0.4 0.6

time t (units of 7500τ)

0.8 1.0

0.20 0.4 0.6

time t (units of 7500τ)

0.8 1.0

0.20 0.4 0.6

time t (units of 7500τ)

0.8 1.0

0.20 0.4 0.6

time t (units of 7500τ)

0.8 1.0

-30

-28

-26

-24

-22

Va

ttr

/ lip

id (

un

its o

f k

BT

)

-30

-28

-26

-24

-22

Va

ttr

/ lip

id (

un

its o

f k

BT

) 1.65

1.7

1.75

1.8

0.02

0.0

0.04

0.06

0.08

0.1

Ve

lec / lip

id (

un

its o

f k

BT

)

0.2

0.0

0.4

0.6

0.8

1.0V

ele

c / lip

id (

un

its o

f k

BT

)

w / σ = 1.6AAc

FIG. 3. (a) Typical snapshots of the lateral phase separation of a charged vesicle consisting of 2500 charged A-lipids [head,red (dark); tail, green] and 2500 neutral B-lipids [head, yellow (light); tail, cyan] at salt concentrations n0 = 1 M and 100 mM.Attractive interaction parameters are set to wAA

c /σ = 1.7, wBBc /σ = 1.7, and wAB

c /σ = 1.5 for both salt conditions. (b)Attractive and (c) electrostatic repulsive potentials Vattr and Velec, respectively, for t = 1.0 × 7500τ under various attractiveinteractions between A-lipids of wAA

c /σ = 1.6, 1.65, 1.7, 1.75, and 1.8 at n0 = 1 M. (d) Attractive and (e) electrostatic repulsivepotentials at n0 = 100 mM. Calculations were performed three times for each n0 and wAA

c /σ value to ensure reproducibility.

Fig. 4(b) clearly indicate the transition point of wABc ,

which coincides with the constant interaction potentialover t = 1.0 × 7500τ . The plateau in Vattr occurred atwAB

c /σ = 1.6 for neutral vesicles, but at wABc /σ = 1.55

for charged vesicles at n0 = 100 mM. This inhibition ofthe lateral phase separation by the electrostatic repulsionamong charged lipids qualitatively agrees with previousexperiments [14–18] and theoretical calculations based onthe free energy of charged membranes [15, 23].

B. Coarsening dynamics

The dynamics of phase separation in lipid vesicles hasbeen extensively studied experimentally and theoretically

in terms of a unified view of scaling behaviors of the do-main coarsening, where the domain size R(t) is propor-tional to tα [13, 52] and α is the domain-growth exponent.Thus, we analyzed R(t) for neutral and charged lipidvesicles. To define the typical domain size, the densitycorrelation function averaged over the vesicle was calcu-lated. Figure 5 shows the spatial distance at which thedensity autocorrelation function of A-lipids (A-A) inter-sects with the density cross-correlation function betweenA- and B-lipids (A-B); this distance is defined here as thetypical size of A-lipid-rich domains. Figure 6 shows thedifferent coarsening dynamics in the neutral lipid vesi-cles, charged lipid vesicles for n0 = 1 M, and chargedlipid vesicles for n0 = 100 mM at the lipid compositionsof A:B = 2500:2500 (Fig. 6(a)) and 1500:3500 (Fig. 6(b)).

6

Neutral

Neutral

1.5 1.525 1.55 1.575 1.6

w /σABc

Charged (n0 = 100 mM)

Charged (n0 = 100 mM)

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1.5251.551.5751.6

Va

ttr

/ lip

id (

un

its o

f k

BT

) w / σ = 1.5ABc

-28

-27

-26

-25

1.5251.551.5751.6

0.20 0.4 0.6

time t (units of 7500τ)

0.8 1.0

Va

ttr

/ lip

id (

un

its o

f k

BT

)

w / σ = 1.5ABc

(a)

(b)

FIG. 4. (a) Typical snapshots of the lateral phase separationof neutral (upper) and charged (at n0 = 100 mM, lower) vesi-cles for wAB

c /σ = 1.5, 1.525, 1.55, 1.575, and 1.6. Other condi-tions are set to wAA

c /σ = wBBc /σ = 1.7, A:B = 2500:2500. (b)

Corresponding attractive potentials Vattr. Calculations wereperformed three times for each charge condition and wAB

c /σvalue to ensure reproducibility.

At A:B = 2500:2500, the domain-growth exponents ofneutral A-lipid-rich domains and charged A-lipid-rich do-mains at n0 = 1 M were both α ∼ 0.5 (Fig. 6(a)),which is generally seen in the coarsening process causedby domain-area deformation driven by line tension [11–13]. Under these conditions, the observed phase sepa-ration occurred by spinodal decomposition with perco-lated domain. This bicontinuous structure with a cleardomain edge was deformed over time by the line ten-sion between the A- and B-lipid-rich phases, which orig-inated from the different attractive interactions betweenthe same and different species. In contrast, the coarsen-

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8 10 12 14 16 18 20

De

nsity c

orr

ela

tio

n

Distance / σ

Typical domain size

all

A-A

B-B

A-B

FIG. 5. Density correlation functions between lipid species.The intersection between the (A-A) correlation curve and (A-B) correlation curve is the typical domain size of A-lipid-richdomains. Typical conditions, where wAA

c /σ = wBBc /σ = 1.7,

wABc /σ = 1.5, A:B=1500:3500, and t = 1.0 × 7500τ , are used

as an example.

ing of the charged A-lipid-rich domains at n0 = 100 mMshowed slower dynamics and α ∼ 1/3, which is generallyseen during evaporation condensation (Ostwald ripen-ing), where the growth exponent of 1/3 is referred toas the Lifshitz–Slyozov–Wagner model [53, 54], or dur-ing diffusion-limited collision and coalescence of smallliquid domains [52]. Figure 3(a) at n0 = 100 mM ex-emplifies the different slow coarsening dynamics in thebinary vesicles in the presence of relatively strong long-range repulsion. The domain edge was blurred in thiscase, and a number of small domains randomly came to-gether rather than a single percolated domain undergoingtension-driven deformation .

In addition, we examined the lipid composition of A:B= 1500:3500 (Fig. 6(b)), for which the area fraction ofcharged A-lipids was almost 0.3, because the coarseningexponent α depends on the area fraction of the phase-separated domains [13]. The coarsening dynamics ofthe neutral lipid vesicles and charged lipid vesicles atn0 = 1 M were again similar. After the initial domainformation with a domain size of more than 2 or 3 lipids,the domain-growth exponent was α ∼ 0.5 as in the caseof A:B = 2500:2500; however, α decreased to ∼ 0.3 orslightly less, probably due to the small number of A-lipidsand A-lipid-rich domains. The small number of lipidsformed small circular domains rather than a percolateddomain, resulting in the collision and coalescence of theliquid domains with an exponent of 1/3. The coarseningdynamics of charged lipid vesicles at n0 = 100 mM werealso an order of magnitude slower than those under neu-tral or well-screened charged conditions. Interestingly,the vesicles remained for a time during the initial homo-geneous state and then suddenly started coarsening, withan ill-defined exponent of α ∼ 0.4 or less.

7

0.1 0.01 1.0

time t (units of 7500τ)

1

0.1

10

do

ma

in s

ize

R/σ

~ t 0.5

~ t 0.333

~ t 0.4

n0 = 100 mMn0 = 1 Mneutral

1

10

do

ma

in s

ize

R/σ

neutral

(a)

(b)

charged (n0 = 1 M)

charged (n0 = 100 mM)

0.10.01

time t (units of 7500τ)

1.0

~ t 0.333

~ t 0.5

FIG. 6. (a) Time evolution of domain size for neutral vesiclesconsisting of neutral A- and B-lipids (black filled circles) andcharged vesicles consisting of charged A-lipids and neutral B-lipids at n0 = 1 M (purple filled squares) and n0 = 100,mM(red open squares). Lipid composition of A:B = 2500:2500.Error bars represent the standard deviation at each time. Cal-culations were performed five times for neutral and chargedlipids at n0 = 1 M, and three times for charged lipids atn0 = 100 mM. Slopes for t0.5 and t0.333 in the double logarith-mic axis are shown as visual guides. (b) Lipid compositionof A:B = 1500:3500 in the same manner in (a). Calculationswere performed twenty times for neutral, and ten times forcharged lipids at n0 = 1 M and 100 mM. Slopes for t0.5, t0.4,and t0.333 are shown as visual guides.

C. Morphological changes

In this section, we turn our focus to the morphologi-cal dynamics of the charged lipid vesicles. In the previ-ous section, we calculated the phase-separation dynam-ics on a timescale of t = 1.0 × 7500τ . However, whenwe continued the calculation until t = 10.0 × 7500τ , weobserved the morphological changes in the charged lipidvesicles depending on the salt concentration n0. In con-trast, we did not observe morphological changes withint = 10.0 × 7500τ for neutral lipid vesicles (data notshown). Figure 7(a) shows typical snapshots of the mor-phological changes, including the topological change from

t = 4.0t = 3.0t = 2.0t = 1.0t = 0.1 (units of 7500τ)

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2.0

Pore

formation

0 4.0 6.0

time t (units of 7500τ)

8.0 10.0

2.00 4.0 6.0

time t (units of 7500τ)

8.0 10.0

Va

ttr

/ lip

id (

un

its o

f k

BT

)

0.48

0.56

0.64

0.72

0.8

Ve

lec / lip

id (

un

its o

f k

BT

)

(c)

(b)

Pore

formation

(a)

FIG. 7. (a) Typical snapshots of the morphological changesof a binary charged vesicle composed of 2500 charged A-lipids(head, red (dark); tail, green) and 2500 neutral B-lipids (head,yellow (light); tail, cyan) at n0 = 100 mM. wAA

c /σ = 1.75,wBB

c /σ = 1.7, and wABc /σ = 1.5. The upper and lower image

series respectively show surface and cross-sectional images ofthe same vesicle. Typically, the morphological changes occuran order of magnitude slower (t ∼ 1.0×7500τ) than the lateralphase separation (t ∼ 0.1×7500τ). Time evolution of the (b)attractive and (c) electrostatic repulsive potentials Vattr andVelec per lipid molecule, respectively. The calculation wasperformed once and showed plausible results (see Fig. 8).

spherical to disk-shaped, which were observed during thelonger time calculation at n0 = 100 mM. The lipid com-position was set to A:B = 2500:2500, and the interactionparameters were set to wAA

c /σ = 1.75, wBBc /σ = 1.7,

and wABc /σ = 1.5. After the relaxation of the lat-

eral phase separation for t = 1.0 × 7500τ (second im-age in Fig. 7(a)), a pore was formed at a charged-lipid-rich domain (red or dark) in the membrane at around

8

t = 1.3× 7500τ . The pore diameter increased with timeand finally the vesicle transformed into a disk shape. Thecharged lipids were segregated at the edge of the disk,whereas the neutral lipids formed a planar bilayer mem-brane surrounded by the charged lipids (last image inFig. 7(a)). During the pore formation, the almost re-laxed attractive potential began to increase (Fig. 7(b)),whereas the increased electrostatic repulsive potentialstarted to decrease (Fig. 7(c)). After the onset of poreformation at t ∼ 1.3×7500τ , these potentials changed asthe vesicle morphology relaxed to the flat disk shape int ∼ 7.0×7500τ . Although the potentials slightly changedeven after t ∼ 7.0 × 7500τ , due to the gradual dropoutsof lipid molecules from the structure, we confirmed thatthe disk shape was kept up to t = 20.0 × 7500τ . Wethus regarded the morphologies at t = 10.0 × 7500τ asthe static states of the pre-assembled vesicles. Note thatsuch states observed in experiments have been treated aspractically static because of the much slower timescale ofcomplete “melting” of the structure.

(a)

(b)

n0 =

1 M

100 mM

10 mM

1.651.6 1.7 1.75 1.8AAc

w / σ

AAc

w / σ

1 M

n0

1.6 1.65 1.7

vesicle

string

bicelle

disk

1.75 1.8

100 mM

10 mM

FIG. 8. (a) Typical morphologies for various salt condi-tions and attractive interactions between A-lipids. n0 is from10 mM to 1 M and wAA

c /σ is from 1.6 to 1.8. wBBc /σ and

wABc /σ were fixed as 1.7 and 1.5, respectively. (b) Phase di-

agram of the charged vesicle morphologies. We defined fourtypes of morphologies; spherical vesicle (circle), disk (square),string (triangle), and bicelle (cross). The dashed lines are vi-sual guides indicating the phase boundaries.

Finally, we determined the phase diagram of the vesiclemorphology for various salt concentrations and attractiveinteraction parameters between A-lipids (Fig. 8). Thesalt concentration, n0, was varied from 1 M to 10 mM,ranging from almost neutral conditions with a corre-sponding Debye screening length of 0.43σ (3.05 A) tohighly repulsive conditions with 4.3σ (30.5 A). The at-traction parameter for A-lipids, wAA

c /σ, was varied from1.6 (liquid state) to 1.8 (gel state). In this wide param-eter region, Fig. 8(a) shows the typical morphologies ofcharged lipid vesicles coupled to the lateral phase sep-aration observed at t = 10.0 × 7500τ , when the vesi-cles reach static states as shown in Figs. 7(b) and 7(c).At n0 = 1 M, the lipid vesicles remained spherical andmacro-phase separation was observed for all the valuesof wAA

c . At n0 = 100 mM, the vesicles exhibited vari-ous morphologies depending on wAA

c . For wAAc /σ = 1.6,

the initially spherical vesicles became string-shaped. ForwAA

c /σ = 1.65, 1.7, and 1.75, the vesicles became disk-shaped, and for wAA

c /σ = 1.8, they remained spheri-cal. At n0 = 10 mM, where the electrostatic repulsionwas substantially stronger than the attractive interac-tion among lipid molecules, the self-assembled structureswere no longer maintained. From wAA

c /σ = 1.6 to 1.7or 1.75, the vesicles broke into small aggregations, calledbicelles, whereas at wAA

c /σ = 1.8 the vesicles adopted aspiky string morphology. When the transitions to disk,string, or bicelle occurred, the charged lipids assembledat the edge of these static structures. Figure 8(b) showsa phase diagram summarizing these static shapes.

IV. DISCUSSION

In this study, we investigated the effects of screened butstill long-range electrostatic repulsion on the dynamicbehaviors of lipid bilayer vesicles, especially the lateral-phase-separation dynamics and morphological changes.After a temperature quench below the miscibility transi-tion temperature, the phase-separation dynamics of a ho-mogeneous neutral lipid membrane are nucleation growthfrom a metastable state or spinodal decomposition froman unstable state. During the nucleation process, thereis an effective energy barrier against the localization of alipid species, owing to factors such as the mixing entropyof lipids and the line tension of domains. In contrast, dur-ing spinodal decomposition, a binary lipid mixture imme-diately segregates into two coexisting phases. The occur-rence of these two instability processes generally dependson the level of the parameter quench at the onset of thephase separation and the lipid composition of the vesi-cle, namely, whether the system state is located below thespinodal line or between the binodal and spinodal linesin the two-parameter field [13, 55]. For example, Veatchand Keller reported the instability of stripe-shaped do-mains with only neutral lipids, suggesting spinodal de-composition in a vesicle of (dioleoylphosphatidylcholine;DOPC):(dipalmitoylphosphatidylcholine; DPPC) (1:1)

9

and 35% cholesterol caused by a temperature quench nearthe critical miscibility temperature [9]. After the onsetof the phase separation by nucleation or spinodal decom-position, the emergent domains grow further by evapo-ration condensation (Ostwalt ripening), collision and co-alescence by diffusion, and domain deformation by linetension, depending on the area fraction of the domain-forming lipids [13]. These domain growth processes atthis stage are called “coarsening”.

The present numerical calculations revealed that theattraction parameters, the lipid composition, and theelectrostatic long-range repulsion play important rolesin the coarsening process and the membrane instabil-ity. Although it is difficult to observe the exact on-set of the phase separation owing to the short lengthand fast timescale, our MD simulation can shed lighton this phenomenon. In the coarsening dynamics ofcharged lipid membranes at n0 = 100 mM for the lipidcomposition A:B = 2500:2500, the lateral structure ofthe membrane remained homogeneous for a certain du-ration, whereas the neutral lipid membranes or chargedlipid membranes at n0 = 1 M immediately began thephase separation (Fig. 6(b)), even though the parame-ter conditions were the same. This result strongly indi-cates that the dynamics of the charged lipid membranesat the onset of the phase separation are not spinodaldecomposition, but instead nucleation growth startingfrom a metastable state with an energy barrier, prob-ably originating from the electrostatic repulsive forceamong charged lipids, that needs an overcoming time t ∼0.1 × 7500τ . During late phase separation, the domaingrowth in charged membranes was much smaller thanthat in neutral membranes (Fig. 6), and growth exponentα values were also different from those in neutral mem-branes. Pataraia et al. previously observed transientsmall circular domains or bicontinuous domains even af-ter 20 min from vesicle preparation of ternary chargedvesicles composed of negatively charged dioleoylphos-phatidylglycerol (DOPG(−)), egg sphingomyelin (eSM),and cholesterol, indicating the slower coarsening dynam-ics. This finding agrees well with our numerical calcu-lations and implies the underlying general physicochem-ical effects on the phase-separation process of electro-static interaction among charged lipids. However, thedetails of the coarsening dynamics, such as growth ex-ponent α of charged lipid membranes, remains less wellunderstood. In this study, we observed an ill-defined ex-ponent α ∼ 0.4 for A:B = 1500:3500 [Fig. 6(b)]. Two-dimensional charged membranes can be exempted fromelectroneutrality because their counterions can escapeinto the three-dimensional bulk, and so the coarseningdynamics in charged membranes might be a unique fea-ture of a two-dimensional electrolyte solution. Therefore,examining α for charged lipid membranes in other ex-perimental or theoretical works would be an interestingstep toward understanding the dynamic heterogeneity incharged membrane systems.

Li et al. recently reported the first systematic theo-

retical study of the equilibrium conformational changescaused by the electrostatic interactions among surfacelipids by calculating the minimum bending and Coulom-bic energies [56]. Although they assumed a fixed uni-form surface charge distribution, our model can treat amore plausible situation, in which the discrete chargedand neutral lipid mixtures move laterally in the mem-brane. Thus, we found the characteristic topologicalchange caused by pore formation in a charged membrane(see Fig. 8). In addition, our MD simulation visualizeddetails of the behavior at a molecular level. Figure 9shows an example of the dynamics of lipid moleculesaround the pore-forming site within the charged-lipid-rich domain (red or dark) during the topological change.Interestingly, even before the onset of the pore formationat t = 7.8 × 7500τ , the orientation of several chargedlipids became perpendicular to that of the surroundinglipids at t = 7.4 × 7500τ . The charged head group ofthe perpendicular lipid was embedded between the twoleaflets of the bilayer membrane, probably because of thestrong electrostatic repulsion among their charged headgroups. This could trigger further instability in the mem-brane structure by interfering with the hydrophobic or-dering attraction of the surrounding lipid chains, result-ing in the pore formation.

It should be noted that in the present calculation, thevesicle diameter was on the order of 10 nm due to the lim-itation imposed by computational cost. Previous studieshave also performed similarly sized coarse-grained simu-lations to reproduce the dynamics of neutral lipid bilayervesicles consisting of, for example, 4000–16000 lipids [49],4096 lipids [45], and 5072 lipids [46]. The curvatureeffect, arising from the nature of the bilayer, in thesetypical coarse-grained calculations is larger than that ofGUVs with negligible curvature, and the vesicles preferto flatten, creating a pore. Because of the curvature ef-fect, the values for GUVs can vary compared with thosereported here. However, the magnitude of the bendingenergy per lipid molecule of ∼ 0.02kBT , estimated fromtypical bilayer bending stiffness κ ∼ 10kBT and curva-ture radius ∼ 10 nm, is much smaller than that of the at-tractive interaction potential of about −20kBT and thatof the electrostatic interaction potential of ∼ 0.8kBT forn0 = 100 mM. Therefore, the deviation from the valuesfor GUVs should be small, and the obtained values hereshould be relevant to GUVs and biomembranes on themicroscale.

The consistency of the topological changes observed inthis study with our previous observations of characteris-tic pore formation in charged lipid vesicles [44] validatesthe plausibility of our coarse-grained MD simulation ofcharged lipid vesicles. The critical salt concentration forthe charge-induced pore formation in charged lipid vesi-cles was determined to be on the order of 100 mM, whichis the typical salt concentration to which biomembranesare exposed. In other words, this salt concentration levelis necessary to maintain the closed membrane morphol-ogy. Considering that biomembranes contain only neg-

10

t = 8.6 t = 9.0 t = 9.4 t = 9.8t = 8.2t = 7.8

Onset ofpore formation

Beforepore formation

Afterpore formation

t = 7.4t = 7.0(units of 7500τ)

t = 8.2t = 7.8t = 7.4

t = 8.2t = 7.8t = 7.4

FIG. 9. Example of the molecular mechanism of pore formation. The vesicle is composed of 2500 charged A-lipids (head,red (dark); tail, green) and 2500 neutral B-lipids (head, yellow (light); tail, cyan) at n0 = 100 mM. wAA

c /σ = wBBc /σ = 1.7

and wABc /σ = 1.5. Sequential snapshots in the middle show the surface and cross-sectional images taken at t = 7.0 × 7500τ to

9.8× 7500τ . Dotted squares contain magnified images of the pore-forming site at t = 7.4× 7500τ (before), 7.8× 7500τ (onset),and 8.2 × 7500τ (after), for surface (upper) and cross-sectional (lower) views of the bilayer membrane. Embedded lipids andthe surrounding lipids are displayed with spheres of the original size and small spheres, respectively.

atively charged lipids and thus are subjected to electro-static repulsion, the topological and morphological sta-bilities of biomembranes could also be physically main-tained by the screened electrostatic interactions at a saltconcentration of ∼ 100 mM. Moreover, the morpholog-ical dynamics of biomembranes, such as exo- and endo-cytosis, may be finely controlled by the precise tuning ofthe electrostatic repulsive forces via the salt concentra-tion near the critical point. This should be investigatedby future studies, including the effects of lipid asymmetrybetween the two leaflets, other charged proteins, explicitsolvents, and so on, with comparisons to experiments andmore detailed simulations.

V. CONCLUSION

In this study, we focused on the dynamics of bi-nary charged lipid vesicles, which are typical soft ma-terials physiologically relevant to the basic structure ofbiomembranes. The important physicochemical phenom-ena in this system are the lateral phase separation andthe morphological changes. By systematically changingthe interaction parameters and salt concentration, we

found that the lateral phase separation is significantlydelayed and/or inhibited in the presence of electrostaticrepulsion. The analysis of domain size in neutral andcharged lipid vesicles suggested a different mechanism ofphase separation in neutral and charged vesicles, wherethe domain-growth exponents for spinodal decomposi-tion and nucleation growth were observed, respectively.Moreover, the longer term calculation unveiled the mor-phological dynamics, such as pore formation and furthertransformation into disk, string, and bicelle shapes, de-pending on the attractive interaction among lipids andsalt concentration, as summarized in the phase diagram.The MD visualization suggested that the pore formationin the charged-lipid-rich domain was initiated by localdisturbances of the orientational order of bilayer mem-branes, which appears prior to the pore formation. Ourfindings provide fundamental insights into the structure,dynamics, and stability of lipid bilayer membranes andvesicles containing charged lipids, which are broadly seenin biomembranes, as well as artificial membranes mim-icking the functional features of biomembranes.

11

ACKNOWLEDGMENTS

Most of the calculations were performed using the par-allel computer “SGI UV3000” provided by Research Cen-ter for Advanced Computing Infrastructure at JAIST.H.I. acknowledges support from Grants-in-Aid for theJapan Society for the Promotion of Science (JSPS) Re-

search Fellowship for Young Scientists (JP13J01297) andSasakawa Scientific Research Grant from The Japan Sci-ence Society (28-233). N.S. acknowledges support fromthe Grant-in-Aid for Scientific Reserch on Innovative Ar-eas “Molecular Robotics” (JP15H00806) from the Min-istry of Education, Culture, Sports, Science, and Tech-nology of Japan (MEXT) and the Grant-in-Aid for YoungScientist (B) (JP26800222) from JSPS.

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