Analytical model and multiscale simulations of Aβ peptide aggregation in lipid membranes: towards a...

8940 Phys. Chem. Chem. Phys., 2013, 15, 8940--8951 This journal is c the Owner Societies 2013

Cite this: Phys. Chem.Chem.Phys.,2013,15, 8940

Analytical model and multiscale simulations of Abpeptide aggregation in lipid membranes: towards aunifying description of conformational transitions,oligomerization and membrane damage†

Martina Pannuzzo,a Danilo Milardi,b Antonio Raudino,a Mikko Karttunen*c andCarmelo La Rosa*a

The mechanisms underlying the formation of extracellular amyloid plaques on neuronal membranes, a

major hallmark of Alzheimer’s disease, are the subject of intense debate. Here we use multiscale

simulations and analytical theory to unveil the early steps of the spontaneous self-assembly of

membrane-embedded a-helical Ab (1–40) peptides. Based on a simple analytical model describing the

electrostatic repulsions among water-exposed charged residues, the presence of distorted structures

called ‘‘frustrated helices’’ is predicted. Large scale (20 ms) Coarse Grained simulations of 36 replicas of

Ab (1–40) performed within a POPC lipid matrix confirmed the formation of supramolecular assemblies

which resemble a twisted ribbon. Fully atomistic simulations have demonstrated the stability of these

helical structures. Concomitant to the formation of these large assemblies, CG simulations evidenced

membrane curvature and substantiate the view that these assemblies may entail mechanical stress on

membrane structure. We think that these findings provide an alternative view to the traditional models

that consider a conformational transition towards b-sheet rich structures as a prerequisite for triggering

membrane damage and, eventually, neurotoxicity.

1. Introduction

Amyloid fibril depositions on neuronal membranes is a typicalhistological feature of Alzheimer’s disease.1,2 The main compo-nent of these depositions is a b-sheet rich aggregate formed bythe b-amyloid (Ab), an amphiphilic peptide of 39–42 residues.2

The observation of Ab aggregates together with in vitro andin vivo biological evidence led to the proposal of the Ab‘‘amyloid hypothesis’’ of Alzheimer’s disease (AD).3,4 Thishypothesis postulates a pivotal role played by Ab production,assembly and fibrillogenesis in AD pathogenesis. In particular,

high levels of Ab may cause neuronal death by (i) bindingto membrane receptors and interfering with their normalactivity;5 (ii) blocking a number of signaling pathways6–8 and(iii) directly perturbing neuronal membranes stability, leadingto unbalanced metal homeostasis.9

However, in apparent contrast with the amyloid hypothesisthere are significant discrepancies between the amount ofamyloid aggregates in an AD brain and the severity of cognitiveimpairment observed in humans10 and animal models.11,12 Toresolve this inconsistency, the notion of ‘protofibrils’,13–22

small-sized oligomers that are intermediate between Ab mono-mers and fully developed amyloid fibers, has been introduced.These intermediates are thought to be toxic in vitro and in vivo,whereas mature Ab-amyloid fibers are supposed to be largelyinert.23 Despite intensive studies their nature remains ratherelusive as even very small changes in the preparation protocolsmay lead to contradictory results.24 Many differently sizedintermediates have been observed: dimeric and trimeric Aboligomers,25–27 56 kDa assemblies18 and even larger structuresthat consist of 2428 and more Ab peptides.29–34 Notwithstandingthese efforts, how Ab deposition into an ordered aggregate islinked with neurotoxicity is still a matter of intense debate.

a Department of Chemical Sciences, University of Catania, Viale A. Doria 6,

95125 Catania, Italy. E-mail: [email protected]; Fax: +39 95 510838;

Tel: +39 95 7385114b Istituto di Biostrutture e Bioimmagini, CNR, Unita Organizzativa e di Supporto di

Catania, Viale A. Doria 6, 95125 Catania, Italyc Department of Chemistry and Waterloo Institute for Nanotechnology, University of

Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1.

E-mail: [email protected]; Fax: +1-519-746-0435;

Tel: +1-519-888-4567 ext. 31390

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c3cp44539a

Received 15th December 2012,Accepted 22nd March 2013

DOI: 10.1039/c3cp44539a

www.rsc.org/pccp

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PAPER

This journal is c the Owner Societies 2013 Phys. Chem. Chem. Phys., 2013, 15, 8940--8951 8941

It is evident that in order to relate Ab structure to neuronaltoxicity, an atomic level picture of membrane-bound Ab oligo-mers is needed.

Experimental assessment of the atomistic structure of Abassemblies in a membrane is a challenging task because oftheir poor solubility and noncrystalline nature.28 In silicomodeling offers an alternative approach35 and Ab aggregationhas been investigated extensively by computational methodsand in particular molecular dynamics (MD) simulations.36,37

Although most simulations have been performed withouta membrane,38,39 Ab assemblies have been investigated inimplicit membrane models,40 and in a dipalmitoylphos-phatidylcholine (DPPC) membrane, the structure and the inser-tion of a single Ab40 peptide41 and the associated changesof the bilayer42 have been addressed. Even the effect thatcholesterol exerts on aggregation has been recently studied ina microsecond MD simulation.43 It is known that Ab aggrega-tion involves a conversion from a random coil to a b-sheetstructure, and that the kinetics of this conformational transi-tion is increased in the presence of membranes.44 However, it isunclear whether b sheet-rich structures are needed to inducemembrane damage. It has been reported that Ab fibrillogenesisinvolves the presence of short-lived oligomeric a-helical inter-mediates that could, in principle, also induce membranedamage and neuronal death.45 To examine this issue, a replicaexchange MD study of the Ab42 monomer with DPPC anddioleylphosphatidylserine (DOPS) bilayers has been performedpointing out that the interactions between Ab and lipids do notnecessarily implicate an alteration of the peptide conforma-tional dynamics.46 Moreover, recent MD investigation reportedthe aggregation of amyloidogenic proteins as a molecularrecognition process47 and the central role played by electro-static interactions48 and bilayer fluidity49 between Ab and thezwitterionic lipid membrane. However, due to the limitationsin computational resources and the few peptides used, acomplete picture of all the concomitant events accountingfor conformational transition, amyloid formation and mem-brane damage is still far from being accomplished. Atomisticsimulations of proteins are usually limited to the time scale of10–100 ns, which is not enough to simulate peptide aggrega-tion. Accordingly, Coarse Grained (CG) models and multiscalemodeling techniques are viable options.39,50

Here we report a multiscale study of the self-assembly of36 a-helical Ab peptides in a model lipid bilayer palmitoy-oleyl-phosphatidylcoline (POPC) in a microsecond time scale. Weemployed a combination of atomistic MD (to explore theconformation of the assembled proteins) and CG simulations(to unveil the dynamics of protein aggregation in themembrane) and used a mapping procedure to switch betweenthe different levels of representations. In order to gain moreinsight into the evolution and formation mechanisms of pro-tein aggregates, we developed a simple analytical model forlinear self-aggregation of Ab at the membrane–water interface.We show that the electrostatic repulsion among water-exposedcharged residues may lead, above a critical threshold, todistorted structures called ‘‘frustrated helices’’. Several small

helicoidal aggregates observed in our MD simulations supportthe validity of the model.

2. Materials and methods

We performed a combination of CG and atomistic (AA) MDsimulations at 300 and 400 K (see Table 1).

Reverse mapping51 was used to connect the different lengthscales. This protocol allowed for longer simulation times anddetailed studies of the intermediate states. The NMR structureof Ab40 (pdb ID:1BA4) was first energy minimized using themethod of steepest descents. This structure was chosen since itshould provide a good starting point for membrane embeddedsystems. A similar approach has been used for b-adrenergicreceptors.52 Pre-equilibration of 20 ns was done using atomisticMD in the NVT (constant particle number, volume and tem-perature) ensemble. This was followed by an equilibrationsimulation run of 50 ns. All simulations, CG and MD, werecarried out using Gromacs 4 53 software and methodologicalissues54 related to neighborlist updates and charge groups weretaken into account. GROMOS 53A655 force field was used in allatomistic MD as it has been shown to perform well in simula-tions of membranes and peptides/proteins.56,57 In particular,the performance of this force-field was in excellent agreementwhen compared directly with experiments on hairpin structuresand detailed contact formation.53 Coarse-grained models basedon the MARTINI58–60 force field were constructed based on thefinal conformations.

CG simulation was performed by using the MARTINI forcefield. Lipid bilayer, pre-equilibrated (100 ns) 128 lipid POPCmembrane61 hydrated by 1500 water molecules was used as astarting point. This bilayer was replicated to obtain a mem-brane with 1596 lipids, energy minimized by the steepestdescents algorithm (1000 steps), and equilibrated in the NPT(constant particle number, pressure and temperature) ensemblefor an additional 100 ns. Finally, systems consisting of Ab40and a bilayer were constructed. Peptides were inserted insidethe membrane perpendicularly to the membrane surface. TheCG systems were hydrated in the presence of 0.1 M NaCl.Cl� counterions were added to preserve charge neutrality.This system was again energy minimized by the steepestdescents algorithm and equilibrated in the NpT ensemble for100 ns. We used the Berendsen weak coupling thermostat and

Table 1 Simulations were performed at 300 K. The asterisk refers to 400 K. AAstands for all-atom and CG for coarse-grained. The production runs werepreceded by at least 350 ns of equilibration; see the section Materials andmethods for details

SystemLipid/peptideratio

Length of eachproduction run Protocol

Ab40, 9 replicas in POPC 192 : 1 1.5 ms CGAb40, 16 replicas in POPC 100 : 1 1.5 ms CGAb40, 36 replicas in POPC 36 : 1 1.5 ms CGAb40, 9 tetramers in POPC 36 : 1 20 ms CGAb40, 24-mer 36 : 1 300 ns AAAb40, 24-mer* 36 : 1 40 ns AA

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barostat algorithms62 with coupling constants of 0.3 ps and3.0 ps, respectively.

In atomistic MD, the starting configurations were obtainedby extracting a portion of the bilayer containing the structure ofinterest, maintaining the same protein/lipid ratio, from the CGsimulation. After extraction the obtained system was equili-brated for 20 ns. The SPC (simple point charge) water model63

was used. The time step was set to 2 fs and the temperature waskept constant using the Nose–Hoover algorithm64,65 with a timeconstant of 0.1 ps. Periodic boundary conditions were applied.The Parrinello–Rahman algorithm66 was applied for semi-isotropic pressure coupling (1 bar). The particle-mesh Ewald(PME) algorithm67 was used for electrostatics.

Atomistic MD production runs were performed as follows:monomeric Ab40 embedded in a POPC membrane was simu-lated for 40 ns. 24-meric assemblies obtained at the end of20 ms CG simulation were simulated at 300 K for 0.3 ms. Thetemperature was then increased to 400 K for 40 ns.

3. Results and discussion3.1. Self-assembly of Ab (1–40) in a POPC lipid bilayer: helicalcontent and oligomer size

To study the effect of peptide concentration on the mechanismof Ab aggregation in a membrane environment, we first run aset of 1.5 ms CG simulations at lipid/peptide ratios of 192 : 1,100 : 1 and 36 : 1 (see Table 1). These simulations suggest thatthe aggregate size is concentration dependent. In order toanalyze quantitatively the results, Fig. S1 (ESI†) shows the timeevolution of monomers (full lines) and that of the averageaggregate size (dotted lines) for different protein concentra-tions. As expected from classical sequential polymerizationkinetics, monomer decay rate is faster at high protein fractions,while the plot of the average aggregation number vs. timeshows a maximum. The position of the maximum is shiftedto shorter times as the protein concentration increases. Asexpected, upon increasing lipid/peptide ratios the formationof large aggregates is promoted. Importantly, in each case theAb self-assembling process proceeds via a mechanism wheredimers bind to other dimers or trimers. In other words

aggregate growth occurs via aggregation of small-sized entitiesthat are larger than monomers. Moreover, by sampling theaggregate size vs. lifetime (Fig. S2, ESI†), the tetrameric stateappears to be the most populated in all simulations at differentlipid/peptide ratios. Once formed, these Ab tetramers diffusedas units and peptides did not dissociate from them.

The CG approach provides information about peptide andmembrane dynamics, but not about the conformational pre-ferences of membrane-embedded Ab peptides. To address thisissue, we performed AA MD simulations of membrane-embedded monomeric and dimeric Ab assemblies. Helicalstructures of monomeric Ab unfold after 40 ns of AA simula-tions leaving only a residual alpha helix encompassing residuesfrom Gly25 to Ala30 (see Fig. S3, ESI†). In contrast, dimersremain in alpha-helical conformations (Fig. S3 and S4, ESI†)even after 250 ns.

To better substantiate our GC strategy, we also performed2 ms simulations of Ab peptide inserted into a lipid matrixmade of negatively charged POPG (10%) and zwitterionicPOPC (90%). These simulations show that in the presence ofnegatively charged lipids Ab peptides show a tendency to lay onthe membrane surface, in full accordance with experimentaland simulations68,69 data and represent a further validation ofresults obtained with pure POPC.

3.2. Self-assembly of Ab40 induces membrane curvaturedepending on the Ab aggregate size

The above simulations (CG 36 replicas) demonstrated thattetramers are the most stable oligomers during aggregation.Once formed, they do not release Ab units and coalesce toform larger aggregates. Based on these data it is conceivable toconsider tetramers as the ‘‘building blocks’’ of Ab self-assembly.Therefore, 20 ms CG simulations of 9 tetramers (36 replicas) ofmembrane-embedded Ab peptide were performed at 300 K. Arepresentative snapshot of this simulation is shown in Fig. 1a.

After 20 ms three kinds of aggregates containing 24, 8 and4 Ab molecules were present. By observing the time evolutionfrom Fig. 2, it can be seen that at 5.2 ms, membrane curvatureis clearly increased. To provide a quantitative description ofthis phenomenon, we introduce two parameters, the radius of

Fig. 1 CG simulation of 36 Ab40 molecules within the POPC bilayer. Frame: a (0 ms), b (4.5 ms), c (5.7 ms), d (20 ms).

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curvature and mean curvature (Mc) to account for size of Abassemblies and membrane curvature respectively.

The radius of gyration Rg is defined as:

Rg2 ¼

Pi mi ~ri �~rcð ÞP

i mi(1)

where mi is the mass of i-th atom, ri is the 3D coordinate of thei-th atom and rc the 3D coordinate of protein center of mass.Mean curvature (Mc) of a surface is defined by

2Mc ¼ 1

Rxþ 1

Ry(2)

where Rx and Ry are the curvature radii defined along two ortho-

gonal axes in a local coordinate frame:1

Rx¼ @2B=@x2

ð1þ ð@B=@xÞ2Þ3=2and

1

Ry¼ @2B=@y2

ð1þ ð@B=@yÞ2Þ3=2, where B is the local deformation from

planarity.The values of curvature radii and gyration radius vs. time are

shown in Fig. 2.From Fig. 2 it is evident that aggregation is a highly coopera-

tive process and membrane curvature occurs consequently to theformation of large (24-mers) aggregates. Moreover, membranecurvature increases occur in two steps: the first at about 5.2 msand the second at about 15 ms. There is also a concomitantincrease in radius of gyration at 5.2 ms and a decrease at 15 ms.This seemingly strange behavior is consistent with the structureof the different aggregates and it will be discussed shortly.

We discuss the Ab 24-mer aggregate further, since experi-mental results70 and MD simulations71 indicate this aggregateto be a candidate to be the first cause for the formation of theaspecific ion channel in lipid membranes. To study this inmore detail, we performed AA MD simulations by reversemapping of the CG coordinates obtained at the end of thesimulations into atomistic ones.

Fig. 3 shows a 24-mer Ab aggregate in a POPC membraneafter 300 ns of AA MD simulation performed at 300 K. Abmolecules in the aggregate remain in a-helical conformationsand show local ordering (inset of Fig. 3) by dimeric or trimericunits with chiral structure. The angle formed between twistedhelices was 39 � 5 degrees.

AA MD simulations of the 24-meric system at 300 K did notreveal any tendency for the peptide to adopt b-sheet structures.These are traditionally considered the precursors of amyloidformation in solution, but not necessarily of oligomers inter-acting with membranes.37,72,73

24-mer Ab aggregates observed in our simulations hadcone-shaped cross-sections, mainly due to the electrostaticrepulsions between charges localized in the water-exposedregion of the outer monolayer (Fig. 4).

Fig. 2 Time evolutions of the radius of gyration (curve blue) and membranecurvature (curve red) as defined by eqn (1) and (2) respectively, for Ab40assembly within the POPC membrane obtained from 20 ms of CG simulations.

Fig. 3 24-mer Ab aggregate inside a POPC membrane after 300 ns of AA MDsimulation performed at 300 K. Ab40 molecules in the aggregate remain ina-helix conformations and show internal organization demonstrated by dimericand trimeric units with chiral structure. For clarity, only one aggregate is shown inthe lower part of the figure. The inset represents a snapshot of the twisted helix.The angle formed between twisted helices was 39 � 5 degrees.

Fig. 4 Cartoon representation of a 24-mer Ab aggregate inside a POPCmembrane. The aggregate has a cone-shaped cross-section and looks ellipsoidalfrom the above. The ellipsoid major and minor axes were a = 92 � 4, b = 45 � 2,a0 = 75 � 3, b0 = 53 � 2 Å, respectively.

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To speed up the dynamics, we increased the temperature inthe AA MD simulations to 400 K. The 24-mer Ab assemblyshows significant curvature stress induced by the aggregate asdisplayed in Fig. 5. These data further confirm the observationsreported in Fig. 3.

Considering that Ab have six negatively charged residues(D1, E3, D7, E11, E22, D23) and three positively chargedresidues (R5, K16, K28), yielding a net charge of �3, theobserved conical geometry can be explained by consideringthe electrostatic repulsions of the charged residues on the sameside of the bilayer, while a contrasting effect is exerted by thelipid matrix that tends to line-up the peptides perpendicular tothe membrane plane. We are now in a position to explain thebehavior shown in Fig. 3 and 5. Peptide aggregation firstincreases the gyration radius of the assembly. When theaggregate becomes more compact, Rg decreases. At the sametime, aggregation facilitates membrane curvature as observedin Fig. 3 in the case of loose aggregates. As the final proteinassembly becomes tighter, its structure assumes a compactwedge, or conical, shape (Fig. 4) mainly because of the above-described torsion deformations of the a helices. This geometryleads to a considerable bending of the lipid membrane asclearly shown in Fig. 3 and 5. The coupling between membranecurvature and torsion of a membrane-buried aggregate will bediscussed in detail in Section 4.3.

We observed an increase in membrane fluctuationsalong the z-axis in the MD simulations of 24 Ab moleculesand the tendency of Ab molecules to go out of the membrane(Fig. 5).

4. Theory of ‘‘frustrated helices’’

One of the most interesting results we have obtained thus far isthe nucleation of distorted structures within the aggregates

presumably originating from minimization of the electrostaticrepulsions. These structures may act as nucleation sites forhelical structures at the membrane–water interface. The evolu-tion and growth of these structures is difficult to observe fortwo reasons:

(a) A computational difficulty is simulating large proteinaggregates in the ms (or longer) time scale.

(b) Simultaneous nucleation of several nuclei due to strongaggregation tendency. This may prevent the formation of longhelices (phenomenon reminiscent of supersaturation andsupercooling behavior in crystal nucleation and growth).

Because of the above constraints, we developed a simpli-fied analysis to extrapolate the geometry of the proteinaggregates over larger time and length scales, as well as toinvestigate their possible influence on the local membranecurvature.

4.1. Energy of a twisted ribbon

Consider a self-aggregate made up of N asymmetrically chargedand elongated macromolecular units (peptides) embedded in aplanar lipid bilayer. They are assumed to aggregate together toform ribbon-like structures. Interactions among identicallycharged residues at the aggregate rim stabilize distorted helicalarrangements owing to the smaller electrostatic repulsion thanin undistorted arrangement (see Fig. 6A). Planar ribbons mayevolve toward either helical or twisted ribbons, their energiesbeing slightly different.74–76 We consider the simplest twistedribbon arrangement where only torsional energy is involved.In isotropic solvents we expect that, upon self-aggregation, theribbons’ rims are identically charged. A different scenario holdswhen we consider a ribbon in a strongly anisotropic mediumsuch as the lipid membrane–water interface. Here, only the rimfacing the aqueous phase bears significant net charges, whilethe opposite rim, deeply embedded in the membrane hydro-phobic region, is basically uncharged (see Fig. 6B). Peptidesoften symmetrically span the entire bilayer. The two polar ends,however, are seldom similar. In most cases (as in the presentstudy) differences are significant. For this reason, we consid-ered only a single charged rim.

Let us summarize the above: while electrostatic repulsionfavours helical deformations in order to maximize the inter-charge distance, in anisotropic systems the same charges stabilizethe ribbon’s planar arrangement by maximizing ribbon–waterinteractions. Competition between these opposite demands givesrise to a peculiar arrangement: the ‘‘frustrated helix’’. We describethe main energy terms below.

The energy of a ribbon-like aggregate can be written as thesum of torsion, electrostatic and membrane–peptide interactions:UTOT = UTORS + UELECTR + UINT. In the simplest continuum picture,the torsional energy of a beam reads

UTORS �1

2a

Z Z

0

C@y@z

� �2dz (3)

where Z is the ribbon length and the positive constant Cmeasures the ribbon’s torsional rigidity.77 In the studied

Fig. 5 Snapshots from AA simulation at 400 K show membrane curvature andthe tendency of Ab molecules to extrude out of the membrane.

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system C increases with the strength of adhesive inter-actions among self-assembled peptides. When ribbon rimsare decorated by charged (or bulky) residues, repulsive inter-actions may distort the planar shape. Let F(xnn0) be thepotential between residues located at n and n0, the repulsiveinteraction is

UELECTR ¼1

2

XNn¼1

XNn0¼1ðFðRð1Þnn0 Þ þ FðRð2Þnn0 ÞÞ (4)

R(1)nn0 and R(2)

nn0 being the distances between two residues belongingeither to the same rim or to opposite ribbons’ rims, respec-tively. In isotropic solvents, rims are identically charged bysymmetry. At the membrane–water interface only the oneexposed to water is charged (see Fig. 6B). In the latter case,F(R(2)

nn0) = 0. Since repulsion decays rapidly with distance, inter-actions are considered to be limited to the nearest neighbours.

Elementary geometry given in Fig. 7 shows that Rð1Þn;n�1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2 þ b2 sin2 12ðyn � yn�1Þ

� �qwhere b is the ribbon width and a

the distance between two nearby residues. Analogously, the distanceR(2)

n,n�1 between residues belonging to opposite rims reads as

Rð2Þn;n�1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2 þ b2 sin2ð1

2yn � yn�1ð ÞÞ

q. On passing to a conti-

nuum picture by changing the discrete variable n into a con-tinuum one, y/a, we may approximate: yn � yn�1 E � a(qy/qz).Inserting the above expressions into eqn (4), developing F(R(i)

nn0)

Fig. 6 (A) Schematic drawing of possible shape evolution of a planar ribbon within an isotropic solvent. (B) Schematic drawing of a planar ribbon imbedded in alipid membrane.

Fig. 7 Geometric parameters used to calculate the repulsion energy amongrepulsive residues at the ribbon rims.

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in power series of (qy/qz) and replacing the sums by integrals,we find

UELECTR � UoELECTR þ

1

2a

Z Z

0

�A @y@z

� �2þB @y

@z

� �4" #

dz (5)

A and B being positive constants depending on the strength andthe decay rate of the intermolecular repulsion (see Section 5).

Lastly, we add an energy term, UINT, describing the peptide–membrane interactions. Since peptides are charged, UINT

must show a minimum when the peptide’s charged endinteracts with water. The simplest function satisfying thisrequirement is

UINT �1

2a

Z Z

0

Uoð1� cos yÞ½ �dz (6)

where Uo is the flip-flop energy. Uo can be partitioned into anelectrostatic and an elastic term: Uo = uELECTR + uELASTIC. Thefirst term depends on the hydration energy of peptide’s ionicresidues. Since positive and negative residues have similarsolvation energies, uELECTR p (n+ + n�). The second term,uELASTIC, depends on the lipids’ lateral displacements uponpeptide’s rotation. uELASTIC should be large at low temperatureswhere lipids are packed into an ordered solid-like (gel) struc-ture, dropping at high temperatures where lipids assume afluid-like behaviour. Experiments show that the membraneelastic properties decrease (about one order of magnitude)upon the order–disorder transition.78–80

4.2. Deriving the equation of motion

The total energy is the sum of torsion, eqn (3), electrostatics,eqn (5), and membrane interactions, eqn (6),

UTOT�Uoþ 1

2a

Z Z

0

ðC�AÞ @y@z

� �2þB @y

@z

� �4

�Uoð1� cosyÞ" #

dz

(7)

The equation of motion for the evolution of y is obtained by

balancing viscous, @UVISC= @y�

, and mechanical, dUTOT/dy,forces

@UVISC

@ y� ¼ �dUTOT

dy(8)

where y�� dy=dt, and the functional derivative dUTOT/dy is

defined as

dUTOT

dy� @UTOT

@y� @

@z

@UTOT

@ð@y=@zÞ (9)

The viscous dissipation is related to the velocity of themoving body

UVISC ¼1

2a

Z Z

0

GðZÞy�2dz (10)

G(Z) being a friction coefficient related to solvent viscosity Z andribbon geometry.81

Inserting eqn (7) and (10) into eqn (8), we obtain a non-linear partial differential equation

ðC � AÞ þ 6B@y@z

� �2 !@2y@z2�Uo sin y ¼ Gy

�(11)

In order to reduce the number of parameters, we changed variables:

z ¼ 6B

C � Aj j

� �1=2

B (12a)

t ¼ 6BG

ð C � Aj jÞ2t (12b)

This yields a simpler form

�1þ @y@B

� �2 !@2y@B2�H sin y ¼ @y

@t(13)

with H � 6BUo

C � Aj j2� 0. The + sign in eqn (13) applies when

C > A and the – sign when C o A. Eqn (13) depends on a singledimensionless parameter; its solution yields the twisted ribbonprofile variation y = y(z,t) in space and time. Two limiting casesare worth exploring.

(a) Isotropic solvent. We first focus on the stationary(@y/@t = 0) solution valid at infinite time. In isotropic solventsthe last term in eqn (13) (describing the peptide–membraneinteractions) is zero, whence

�1þ @y@B

� �2 !@2y@B2¼ 0 (14)

Eqn (14) is satisfied when the term in the brackets vanishes.Retaining real and positive solutions and coming back to theoriginal variable z through eqn (12a), we obtain y = Oz + C1, withO � (|C � A|/6B)1/2 if A > C and y = C1 if A o C. The integrationconstant C1 is calculated by imposing that at z = 0 the ribbon isperpendicular to the membrane interface: y|z=0 = 0, thus C1 = 0.Summarizing,

y ¼ 0 C4A

y ¼ Oz CoA

((15)

that is, a transition from planar to twisted ribbon configurationoccurs when the electrostatic repulsion A overcomes the torsionenergy C. Since the coefficients A and B are proportional to thesquare of the rim electric charge Q (see Section 5), it followsthat O increases with Q, levelling up to a constant value at largeQ’s. The associated parametric equations are

x ¼ ðb=2Þ sin y

y ¼ ðb=2Þ cos y

y ¼ Oz

8>>><>>>:

(16)

They describe a regular helix of radius b/2 and constant pitchP = 2p/O as shown in Fig. 6A.

(b) Anisotropic solvent (membrane). When the peptidesaggregate is deeply embedded in the lipid matrix, it cannot

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experience large deformations from planarity. Whencesin y E y and �1 + (@y/@z)2 E �1, in this limit eqn (13)reduces to

�@2y@B2� H y (17)

By imposing the simplest boundary conditions, y|z=0 = y|z=Z = 0(clamped ends), solution to eqn (17) shows that the planarribbon, yeq = 0, is the equilibrium configuration. Next, weinvestigate the torsion fluctuations around equilibrium. Welook for a Fourier series expansion of the torsion fluctuations:

y�yeq =Pq

yq sin qzð Þ, where yq is the unknown amplitude

and q r 1 the wave vector. Inserting the above expansion intothe total energy, eqn (7), averaging over the ribbon length andequating the averaged energy to the thermal energy, we obtainup to quadratic terms

yq2 �kT

ðC � AÞq2 þUo(18)

eqn (18) is meaningful only when

(C � A)q2 + Uo > 0 (19)

as otherwise the fluctuation amplitudes diverge to infinity.When the inequality (19) is satisfied, the equilibrium configu-ration is a planar ribbon, otherwise the system evolves toward adistorted arrangement described by eqn (13).

Apart from the above limiting cases, solutions to the equa-tion of motion (13) can be obtained only by numerical techni-ques as described in Section 5.

4.3. Effect of the ‘‘frustrated helices’’ on local membranecurvature

When a ribbon embedded in a lipid matrix undergoes atransition from a planar to a frustrated helix arrangement,the surface area occupied by the ribbon increases. If thearea increase is not identical in the two bilayer’s leaflets(e.g., because of partial insertion not symmetrically spanningthe bilayer as shown in Fig. 6B), the differential increment ofthe membrane surface area may have a deep influence on themembrane local curvature. Simple analytical theories based onthe area difference model have been developed over the years;82

an important point is the extreme sensitivity of the membranecurvature to small differential increments of surface area.Looking at Fig. 6A and 6B, it is evident that a transition froma planar ribbon configuration to a frustrated twisted ribbonenlarges the area of the upper leaflet while slightly affecting thelower one. Therefore, we conclude that a twisted ribbon favoursthe curvature of peptide rich domains.

The differential surface area increase DDA = DAouter � DAinner

has been calculated as the area difference between the outerand inner monolayers upon insertion of a single ribbon. For thesake of simplicity we have assumed in Fig. 8A an elliptic cross-section of the ribbon long axis Z (Z is perpendicular to the planeof Fig. 8A). From Fig. 8A it is evident that DAouter and DAinner

both depend on the ribbon’s torsion angle y. Indeed, when apeptide is perpendicularly inserted into the lipid membrane a

rather large hydrophilic portion is exposed to water. A torsionburies portions of the peptide’s hydrophilic region into thelipid matrix, causing a surface area increase DAouter. If peptide’scentre of mass coincides with that of the lipid bilayer, we musthave a symmetrical increase of the surface area of the innermonolayer DAinner, hence DDA = 0. However, if the centres ofmass do not coincide (or if the peptide shape is not symme-trical), DAouter and DAinner may differ to a large extent, leading toDDA a 0. Notice the ribbon’s centre of mass set at a distance hfrom the bilayer symmetry mid-plane. The increase of thesurface area of the outer and inner monolayers is proportionalto the embedded fraction of the peptide. This latter quantitycan be easily calculated by surface integration by assuming ashape for the ribbon cross-section. Some representative resultsare reported in Fig. 8B. The increase of differential surface areaDDA with y is evident, the effect being particularly evident whenthe centre of mass of the ribbon-forming peptide is displacedfrom the bilayer’s centre of mass as well as when the peptide’sshape is strongly elliptical.

Fig. 8 (A) Geometrical parameters of a twisted ribbon cross section imbeddedin a lipid bilayer. The elliptical region (with semi-axes a and b) describes the cross-section of a twisted ribbon with a local torsion angle y. The angle y is modulatedalong the z-axis perpendicular to the image. a and b are the short and long axesof the ribbon elliptical cross section, L the monolayers’ thickness, h the position ofribbon’s centre of mass. (B) Variation of the differential area increase DDA (nm2)between outer and inner monolayers vs. the ribbon’s torsion angle y (rads).Curves have been calculated for different values of the distance h of the ribbon’scentre of mass from that of the lipid bilayer. From the bottom to the top: h = 0.1,0.2, 0.3 and 0.4 nm (see Fig. 8A). Other parameters defined in Fig. 3A are: a = 1,b = 2, L = 1.5 nm.

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Once the differential area increase as a function of the angle yhas been calculated, we must average over the ribbon’s length.It is natural to assume a sinusoidal modulation of y: y EyMAX sin (qZ); looking at Fig. 8B we note that around equili-brium DDA p y2; hence, the average over the whole ribbon

length L leads to: hDDAi / 1

L

R L0 yMAX

2 sin2ðqZÞdZ / 12 yMAX

2,

a result independent of ribbon wavelength q.The emerging message from our qualitative calculations is

an enhanced bilayer curvature upon the maximum ribbon’storsion angle yMAX:

curvature E K1 + K2yMAX2

K1 and K2 being positive constants.

5. Numerical calculations

We assumed that the electrostatic repulsion within a ribbon-like peptide aggregate is described by the screened Coulomb

potential FðRðiÞnn0 Þ ¼ Q2e�kRðiÞ

n;n�1

eRðiÞn;n�1, where Q is the residue’s net

charge, e the solvent dielectric constant and k is a measure ofthe decay rate of the repulsive forces (k increases as the squareroot of salt concentration). Power series expansion of theelectrostatic energy yields the parameters A and B defined byeqn (5). For symmetrically charged ribbons, we find

ASymm ¼Q2b2

16eae�kað1þ kaÞ þ 1

x2e�kaxð1þ kaxÞ

� �(20a)

BSymm¼Q2b4

128ea

e�kaðð1þkaÞð52þ2ða=bÞ2Þþ 12k

2a2Þþ

1

x5e�kaxðð1þkaxÞð1þ 3

2xþ2ða=bÞ2x3Þþ 1

2k2a2x2Þ

2664

3775

(20b)

where x � (1 + (b/a)2)1/2, a being the shortest distance amongthe peptides and b the ribbon’s width (see Fig. 6A). An analo-gous calculation for a ribbon with only a single charged rimexposed to water leads to

AAsymm ¼Q2b2

8eae�kað1þ kaÞ½ � (21a)

BAsymm ¼Q2b4

64eae�kað1þ kaÞð52þ 1

2k2a2Þ

(21b)

In deriving eqn (20) and (21), we assumed that symmetricallyand asymmetrically ribbons bear the same net charge.

Setting k E 109 m�1 (a typical value of the Debye screeningat physiological 0.1 M salt concentration), e E 80, a = 10�9 m,b = 4 10�9 m (of order of bilayer thickness), Q = (n+ � n�)e,where n+ � n� is the difference between the number of positiveand negative residues per peptide molecule (we set n+ � n� = 10)and e the unit charge, eqn (21) gives: AAsymm = (n+ � n�)2 10�38 J m2 and BAsymm = (n+ � n�)2 6 10�56 J m4.

The flip-flop energy of activation per single peptide, Uo, wasassumed to be about 70kT, a value larger than that of lipid

molecules (for a lipid surrounded by other lipids at roomtemperature) Ulipid

o /kT E 30–40 (ref. 83) and Ulipido /kT E 50 for

a lipid molecule in cholesterol-containing bilayers.84 Differentresults have been obtained for other amphiphilic molecules.For instance cholesterol, or similar sterols, dispersed in a lipidmatrix exhibits consistently lower values, UChol

o /kT E 3–13.85,86

Fewer and less precise data exist for that concerns peptides, asmall peptide, alamethicin, exhibits flip-flop energies rangingfrom Upeptide

o /kT E 40 (ref. 87) to Upeptideo /kT E 60,88 depending

on the inter-conversion pathway. Moreover, throughout thepaper we assumed that the torsion coefficient C relative to anuncharged planar ribbon was always much smaller that theelectrostatic repulsion among peptides’ charged residues. Inthis limit, results are independent of C. The ribbon’s frictioncoefficient per unit length G(Z) has been approximated to thatof a cylinder. Even after neglecting the variation of viscosity Z atthe interface, formulas are by no means simple. However, if oneis not interested in the kinetics, the detailed knowledge of G(Z)is unnecessary, only a rough value of G(Z) is enough to estimatethe range of the time variable in numerical calculations. We

used the approximation: GTOTðZÞ ¼1

a

RZ0 GðZÞdz � Z

aGðZÞ. Since

the friction coefficient of the whole cylinder reads GTOT(Z) EpZb2Z, we find G(Z) E pZab2. Since membrane viscosity is aboutthree orders of magnitude greater than water viscosity, thefriction coefficient per unit length of a ribbon in a lipid matrixis G(Z) E 10�26 J s. Inserting these parameters into eqn (13) andsolving it by Mathematica 6.0 package, we obtain the 3D plotsshown in Fig. 9. In the plots we report the amplitude of theribbon’s torsion deformations y as a function of the scaledspace, B, and time, t, defined by eqn (12). We started at t = 0from a planar ribbon, y = 0. At later times, the system remainsin its planar state, provided the ribbon’s charge is below acritical threshold (Fig. 9, panel A). On the contrary, above acritical charge density, the ribbon develops periodic deforma-tions (Fig. 9, panels B and C). As expected, deformations arerather small due to the constraints imposed by the lipid matrix.In order to trigger the patterns evolution we added a smallsinusoidal deformation at t = 0 (up to nine orders of magnitudesmaller than that of the final structure), and the results areindependent of amplitude and periodicity of the appliedperturbation.

We checked that the system has reached a stable structurethat does not further evolve with time.

6. Discussion and conclusions

In this work, we have developed an analytical model to thedescribe the Ab structures using ‘‘frustrated helices’’ andperformed multiscale simulations of various Ab aggregates inthe microsecond range. We start by discussing the theoreticalmodel: a theoretical treatment has been developed to interpretthe geometry of the superstructures observed in simulations.Our model demonstrates that these superstructures assume theshape of a ‘‘frustrated helix’’, a geometry arising from the balancebetween electrostatic forces governing membrane–peptide and

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peptide–peptide interactions. Moreover, the formation of thesefrustrated helices induces local membrane curvature. Althoughthe transient nature of these structures escapes the conven-tional experimental investigations, they may represent a criticalearly intermediate state accounting for aggregate growth andmembrane deformation, which are preludes to amyloid deposi-tion. A possible extension of the model would be to includehydrophobicity since hydrophobic interactions are crucial insecondary structure formation.89 The following conclusionscan be made based on the theoretical model:

(a) In isotropic solvents asymmetrically charged elongatedunits self-assemble into ribbon-like structures. To minimizethe electrostatic repulsion the two ribbon’s rims bear identicalcharges. A charged ribbon undergoes a transition from a planarto a twisted arrangement above a critical threshold dictated by thebalance between torsion and electrostatic energies (see eqn (7)).The twist pitch decreases with ribbon’s charge, levelling-up to aconstant value at large charge density.

(b) When a self-assembled ribbon is embedded in a lipidmembrane only the ribbon’s rim exposed to water is charged.Below a critical charge threshold, energy minimization shows thatthe ribbon’s undistorted planar arrangement is the most stableone. However, even in the planar equilibrium configuration,repulsion increases the amplitude of fluctuations (see eqn (18)).

(c) Once the critical threshold has been reached, the systemevolves into patterned structures obtained by numerically

solving eqn (13). Patches spring over time from an initial flatconfiguration (y = 0). The patches contain domains of partiallyright-oriented peptides followed by identically sized patches ofleft-oriented peptides, or vice versa (see Fig. 10).

Two parameters define the topology of the patches: themaximum amplitude of the torsion angle, yMAX, and the wave-length P. Both parameters can be derived from the data of Fig. 9where we report the space and time evolution of the torsionangle y calculated at increasing flip-flopping energy Uo. Upon aproper transformation of variables, it can be seen that patches’wavelength P is of order of 4–5 10�9 m (or 3–5 molecules) forUo = 70kT, increasing to about E15 10�9 m (or 15 molecules)at lower Uo values (Uo = 60kT). Also the times required to attain adistorted configuration are different: the greater the distancefrom equilibrium, the faster the relaxation rate.

(e) Deviations from ribbons’ planarity induces curvature ofthe lipid bilayer. The effect is related to the differential increaseof the surface area of the inner and outer bilayer’s leaflets. Weproved that bilayer’s curvature increases with the maximumamplitude of the ribbon torsion angle.

We also performed CG and MD simulations of up to 36 replicasof membrane embedded Ab40 to simulate their self-assemblyin an environment resembling their physiological milieu in amicrosecond timescale. The simulations revealed the persis-tence of tetrameric building blocks that spontaneously coalesceto assemble into linear aggregates with a structural geometry

Fig. 10 Top (panel A) and lateral (panel B) views of a frustrated helix embedded in a lipid bilayer.

Fig. 9 Time evolution in scaled space (B) and time (t) of the torsion deformation y of a charged ribbon calculated at different values of the flip-flop energy Uo. WhenUo is large (Uo = 80kT) the torsion angle y is always zero (panel A). By decreasing Uo the planar configuration becomes unstable: above a critical threshold permanentdeformations of well-defined amplitude and periodicity begin to develop (Uo = 70kT, panel B). At even smaller Uo values (Uo = 60kT, panel C) the amplitude of thedeformation is wider while patches wavelength is higher.

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reminiscent of a twisted ribbon. Concomitant to aggregates sizegrowth a significant curvature of the lipid membrane isobserved. The formation of large Ab peptide aggregates isobserved only if the peptide/lipid ratio is higher than a thres-hold value of about 1/36. Conversely, at lower peptide/lipidratios the presence of di- and tri-meric assemblies is favored.All-atom MD simulations of the final 24-mer aggregate showedthat the helical secondary structures of single peptides aremaintained over the entire simulation. This finding was alsoconfirmed by fully atomistic MD simulations of monomericand dimeric Ab40 in a membrane environment, evidencing thatassembling of single peptides into two or more units enhancesa-helix stability.

Acknowledgements

We acknowledge the support from Natural Sciences and Engi-neering Research Council of Canada. Computational resourceswere provided by SharcNet (www.sharcnet.ca). We acknowledgeUniversity of Catania for financial support.

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