Twisted Ribbon Aggregates in a Model Peptide System

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Twisted Ribbon Aggregates in a Model Peptide System Axel Rüter, Stefan Kuczera, Darrin J Pochan, and Ulf Olsson Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b03886 • Publication Date (Web): 08 Apr 2019 Downloaded from http://pubs.acs.org on April 8, 2019

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Twisted Ribbon Aggregates in a Model Peptide System ∗,†

Axel Rüter,

Stefan Kuczera,

†

‡

Darrin J. Pochan,

and Ulf Olsson

†

†Division

of Physical Chemistry, Lund University, SE-22100 Lund, Sweden ‡Department of Materials Science and Engineering, University of Delware, Newark, Delaware 19716, United States E-mail: [email protected]

Abstract The model peptides A8 K and A10 K self-assemble in water into ca. 100 nm long ribbon-like aggregates. These structures can be described as β -sheets laminated into a ribbon structure with a constant elliptical cross-sections of 4 by 8 nm, where the longer axis corresponds to a nite number, N ≈ 15, of laminated sheets, and 4 nm corresponds to a stretched peptide length. The ribbon cross-section is strikingly constant and independent of peptide concentration. High contrast transmission electron microscopy (TEM) shows that the ribbons are twisted with a pitch λ ≈ 15 nm. The self-assembly is analyzed within a simple model taking into account the interfacial free energy of the hydrophobic β -sheets and a free energy penalty arising from an increased stretching of hydrogen bonds within the laminated β -sheets, arising from the twist of the ribbons. The model predicts an optimal value N , in agreement with the experimental observations.

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Introduction Self-assembly processes are found throughout nature, where biological molecular building blocks interact with one another and organize into macromolecular structures at a high factor of reproducibility. Obtaining fundamental understanding and control of these structures and their formation has led to an increasing amount of research in the eld of novel biomaterials. 13 Studying the thermodynamics and kinetics of self-assembly can also provide great insight into the world of biology, leading to a better understanding of for example protein folding and misfolding. 4,5 It has specically been shown, that the assembly of proteins or peptides into insoluble one-dimensional brillar structures, are strongly correlated to various diseases such as Alzheimer's and Hungtington's, and in diabetes type II. 68 There are many possible building blocks from which self-assembled peptides bres can be constructed. In the eukaryotic cytoskeleton, globular proteins such as actin and tubulin selfassemble into brillar structures, whereas collagen bres, most often found in brous tissues, consist of intertwined helical collagen proteins. 9,10 The aggregated peptides related to the previously mentioned diseases are however all constructed of β -sheets, laminated to form long twisting insoluble ribbon-like structures. These structures further aggregate forming what is knows as amyloid brils. 11,12 Given the biological signicance of amyloids, many naturally occurring peptides from which they form have been analyzed in great detail. 1315 With the increasing accessibility of synthetically designed peptides, the number of peptide model systems showing similar self-assembly behaviour has increased greatly. 1,16,17 Cenker et al. 18 has systematically studied the self-assembly within the An K model peptide system, consisting of n alanine residues and a single lysine residue at the carboxyl end, by varying the number of alanine residues in the sequence, n = 4, 6, 8, 10. The pentapeptide A4 K does not self-assembly due to a high peptide monomer solubility, A6 K self-assembles into monodisperse peptide nanotubes, and A8 K and A10 K were found to self-assemble into long ribbon-like β -sheet aggregates with a biaxial cross-section of ca. 4 nm x 8 nm. 18 The peptides in the ribbon-like aggregates have an ordered crystalline packing on a 2D oblique 2


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lattice, and can be viewed as a stack of laminated β -sheets, where the β -sheets propagate in the direction of the ribbon. 19 A striking feature of these aggregates is that the cross-section dimensions are constant. 18 There is no observable variation between the two peptides, and furthermore, it does not vary with concentration. A model describing such aggregates, constructed of β -sheet stacks, has been presented by Nyrkova et al., 20,21 focusing on the free energy contributions that dictates the nite cross-section width. The model assumes an attractive interaction between β -sheets leading to aggregation into ribbons, in this paper referred to as lamination. However, due to the peptide chirality the β -sheets also have a spontaneous twist. This twist results in an elastic deformation of the β -sheets giving rise to a free energy penalty for large ribbon widths, and thus stabilizing a nite number of β -sheets in the ribbon. In this paper we analyze the ribbon aggregates in the A8 K and A10 K systems. Following the work of Nyrkova et al. 20,21 we consider an interfacial tension of the β -sheets and a hydrogen bond pair potential. Using high resolution transmission electron microscopy (TEM) the ribbon twist is resolved, from which the β -sheet deformation is estimated.

Experimental The synthetic peptides An K , (n=8,10) were acquired from CPC Scientic Inc. as triuoroacetate stabilized salts with purities > 96% and were used without further purication. Sample concentrations were determined excluding the mass of the counter ion, and both H2 O and D2 O was used as a solvent for comparisons with NMR measurements presented elsewhere. D2 O was purchased from Sigma Aldrich with an isotope purity of 99.9%. Small and wide angle X-ray scattering (SAXS/WAXS) was performed on a Saxslab Ganesha pinhole instrument, JJ X-Ray System Aps, equipped with an X-ray microsource (Xenocs) and a two-dimensional 300k Pilatus detector (Dectris Ltd., Switzerland). The X-ray wavelength was λ = 1.54 Å. Samples were measured at 3 given sample-to-detector distances and

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the scattering intensity was put to absolute scale by calibration against water. Azimuthally averaged intensities as a function of the scattering vector q = (4π/λ)sin{θ/2}, where θ is the scattering angle, were analyzed using inhouse MatLab codes. We assumed that the SAXS intensity, I(q), can be written as

I(q) = φ∆ρ2 hV ihP (q)iS(q) + background

(1)

where φ denotes the peptide volume fraction, ∆ρ is the scattering length dierence between the aggregate and solvent, hV i is the average aggregate volume, hP (q)i is the average particle form factor and S(q) is an eective structure factor. ∆ρ = 1.15 × 10−5 Å−2 for both A8 K and A10 K, and was calculated from their respective chemical composition and densities,

%A8K = 1.50 gcm−3 and %A10K = 1.26 gcm−3 . 18 Samples for scattering measurements were prepared directly to the concentration measured. Negatively stained TEM imaging under ambient temperatures was performed on either a Talos FC200C at the Keck Center for Advanced Microscopy and Microanalysis within the University of Delaware, or a JEOL JEM-2200 using a TVIPS F416 camera at the national Center for High Resolution Electron Microscopy within Lund University. In both cases an accelerator voltage of 200 kV was used. Samples were deposited on ultra-thin formvar and carbon coated copper grids (EMS) and blotted before stained using a phosphotungstic acid (PTA) solution of 2 wt%. Cryo EM imaging was performed on the same JEOL instrument as described above, using the same acceleration voltage. The samples were vitried on lacey carbon lm covered copper grids using a Leica EM GP automatic plunge freezer. Grids used in all EM measurements were glow-discharged for increased wettability before sample preparation and all analyzed samples had been diluted to low enough concentrations to visualize individual peptide aggregates. Image analysis was performed using ImageJ and MatLab.

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Results and Discussion A8K and A10K ribbon structure The short model peptide family An K consists from the N-terminus of n hydrophobic alanine amino acids, anked by a single lysine amino acid at the C-terminus. Apart from the possible charge on the lysine side group, the uncapped (n + 1)-mer also carries potential charges at either terminus leading to a net positive charge in neutral and acidic pH conditions. The peptides self-assemble into ordered β -sheet rich aggregates above a specic concentration, interpreted as the monomer solubility, cs . 18 The solubility decreases strongly with increasing

n due to increased molecular hydrophobicity. For the A8 K and A10 K peptides, the self-assembled aggregates are ribbon-like. 18 SAXS data, presented as scattered intensity, I(q), normalized by the peptide concentration in Figure 1, reveal a number of striking features. The ribbon cross-section is very similar for the two peptides, as seen from the very similar SAXS patterns at higher q -values. Moreover, the cross-section is independent of the peptide concentration as shown by the overlap of the normalized scattering patterns, forming a master curve. Thirdly, the peptide molecules have a crystalline packing in the aggregates as seen by the three diraction peaks in the wide angle regime (q > 1 Å−1 ). 18 The peak positions, marked by vertical lines in Figure 1, are essentially identical for both peptides. The ribbons can be modeled as elliptical rods. 18 The corresponding isotropically averaged form factor is given by 22

2 P (q) = π

Z 0

π/2

Z 0

π/2

2J1 (qr(a, b, φ, α)) sin(qL cos(α)/2) qr(a, b, φ, α) qL cos(α)/2

where r(a, b, φ, α) = a2 sin2 φ + b2 cos2 φ

1/2

2 dφ sin α dα

(2)

sin α, L is the rod length, a and b are the minor

and the major semi axes of the elliptical cross-section, J1 is the rst order Bessel function, φ and α are the angles between the major axis of the ellipse and q , and between the direction of the cylinder and q respectively. 5


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Figure 1: Solution scattering experiments on both A8 K and A10 K respectively. Experimental data has been normalized by concentration and is shown together with model scattering curves for elliptical cylinders. For A10 K an S(q) contribution is present which is calculated within the RPA to extract information at S(0). Dashed vertical lines in the high q region indicate the Bragg reection positions, conrming the same peptide packing within the two peptide ribbons. Model calculations of I(q) are presented in Figure 1 as solid lines. For A8 K, only a single model curve is shown since the normalized data all overlap. Here the cross-section parameters were a = 1.8 nm and b = 3.6 nm. The aggregate length is longer than can be resolved in the experimental q -range where qmin = 5 × 10−3 Å−1 , hence a value of L was chosen only to be L >> 1/qmin . S(q) was assumed to be unity in the present q -range and concentration range as no concentration dependence of the normalized scattering intensity is observed at lower q -values. For the models presented together with the A10 K data, the corresponding cross-section dimensions are a = 2.1 nm and b = 4.2 nm. L is smaller here compared to the A8 K system. For the two lowest concentrations, I(q) levels o at lower

q -values, while at the same time the normalized intensities are the same. This indicates that we approach the Guinier regime at lower q -values and that for these concentrations S(q) ≈ 1 within the present q -range. The model calculations for these concentrations of the A10 K peptides gives L = 60 nm, consistent with cryo-TEM images, discussed below. The A10 K ribbon length is veried through determination of the weight averaged aggregate molecular weight, Magg , of the scattering objects, possible at q → 0 from absolute scale measurements at innite dilution. At these conditions Equation 1 can be rewritten to

6


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I(q → 0) =

cMagg ∆ρ2 NA % 2

(3)

where c is the mass concentration, NA is Avogadro's number, % and ∆ρ are dened as above. The intensity, I(q → 0) is acquired from the intercept of the theoretical scattering curve and the y -axis, shown again together with the appropriate data on absolute scale in Figure 2. Magg is used to determine an average aggregation number of the peptide ribbons,

N = Magg /Mpep = 2045, where Mpep = 857 gmol−1 is the molecular weight of a single peptide monomer. Combined with the structural model presented below, the number density of molecules per nanometer, ρN = 34.1 peptides per nm, is used to calculate the aggregate length L = N /ρN = 60 nm.

Figure 2: The SAXS pattern of 0.5 wt% A10 K together with the theoretical scattering curve of an elliptical cylinder. For the higher concentrations of A10 K, a decrease of the normalized intensity is observed at lower q -values. This is interpreted as an eect of S(q) < 1 in this q -range. A consequence of long range electrostatic interactions. Indeed the system is expected to be repulsive due to the high surface charge of the ribbons, preventing further aggregation into amyloid-like brils. A simple model for S(q) is given by the random phase approximation (RPA), 22

S(q) =

1 (1 + νP (q))

(4)

including an interaction parameter, ν . Although this S(q) does not fully describe the in7


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termediate q regime of the experimental data, an extraction of a reasonable S(0) value is possible. S(0) decreases monotonically with increasing the concentration of A10 K, as shown in Figure 3. The reason why we do not see any structure factor eects for A8 K (up to 3 wt%) is because of the higher monomer solubility cs ≈ 2 mM, 18 corresponding to a Debye length

λD ≈ 7 nm, 23 indicating a strong screening of the long range electrostatic interactions. The monomer solubility of A10 K is much smaller, cs ≈ 8 µM, and does not provide any signicant screening, λD ≈ 100 nm.

Figure 3: S(0) obtained from the modeling of the A10 K scattering data presented as a function of volume fraction. The three Bragg reections observed in the wide angle regime, can be indexed to an oblique 2D lattice. 19 The spacing between laminated β -sheets is 0.54 nm and the β -strand spacing within the β -sheets is 0.45 nm. Using these values the ribbons can be viewed as composed of a nite number of laminated β -sheets, N ≈ 15. Cryogenic visualization using TEM has been performed on A8 K and A10 K respectively and representative images are presented in Figures 4 A and B, where solution structures are conserved by sample vitrication. These images essentially conrm the conclusions drawn from the SAXS data, such as the sti rod-like shapes of the aggregates, the monodisperse width of roughly 8 nm and that the A8 K aggregates are longer compared to those formed by A10 K. 8


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Higher resolution imaging in TEM can be achieved by the addition of a negative staining agent PTA. This is shown in Figures 4 C and D. Even though these samples are analyzed in the dried state, there are no evident dierences between the shape of the observed structures compared to those observed using cryogenic visualization. However, the higher contrast reveals an oscillating intensity pattern along the ribbon lengths. These oscillations are interpreted as the ribbons twisting around their direction of propagation, a common trait in for example amyloid brils. 24 The twist is an inherent feature arising from the chirality of the peptide molecules.

Figure 4: TEM images of A8 K and A10 K respectively in A) and B) using cryogenic visualization and in C) and D) using PTA as a negative staining agent. Scale bars are 100 nm. Further analysis on the twist of the ribbons was performed on randomly oriented ribbons from multiple sample preparations, and only where the ribbons or a major part of the ribbons were determined to not lie on top of anything but the EM grid. Example images of line scans along the ribbons and of the background are shown in Figure 5. Pitch values are obtained by tting a sinusoidal function by eye to the acquired intensity oscillations. The pitch lengths were estimated to 16 nm and 12 nm for A10 K and A8 K respectively, which is roughly half of the optimum pitch found in real proteins. 25 The analysis was performed on 16 and 8 ribbons from A10 K and A8 K respectively. Although we do not observe any dierence in the structures observed in the dry state compared to those in solution, we cannot exclude the possible inuence of the staining and drying conditions on the acquired twist parameters. Other work has shown that the pitch 9


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of twisting ribbons seem to depend on the ionic strength of the solution, 26,27 a parameter which will change with drying and staining.

Figure 5: Individual A8 K and A10 K ribbons in A(i-iii) and D(i-iii) respectively, visualized through TEM measurements using PTA as a negative staining agent. B) and E) show the same ribbons indicating the position of line scans represented in C) and F). Sine wave functions have been adjusted to the line scans to extract the wavelength of the oscillating intensity resulting in half the β -sheet pitch, λ/2. Images denoted BG show line scans of the respective backgrounds. All scale bars are 30 nm. From the available structural data we can construct a three-dimensional theoretical structure model for the ribbon aggregates valid for both the A8 K and the A10 K systems, shown in Figure 6. The ribbons consist of laminated β -sheets at a separation dβ = 0.54 nm. The

β -sheets are antiparallel and constructed of stretched peptide monomer units of length lp , at a separation dp−p = 0.45 nm. Using a smaller spherical cap at the N-terminus and a larger one at the C-terminus the directionality of the peptides is indicated. The β -sheets propagate along the length of the ribbons, twisted with a pitch, λ, of roughly 16 nm. As previously mentioned, the pitch is a hierarchically transferred property determined by the angle θ between adjacent peptides in a β -sheet. From the presented notation the ribbon aggregates consist of N = (2M + 1) β -sheets in total, where M is the maximum value of m in ribbon of nite width. The central β -sheet, m = 0, is only twisted with the angle θ between 10


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adjacent peptides and the midpoints of the stretched peptides lie on a straight line. For

β -sheets where m 6= 0, the midpoints no longer lie on a straight line, instead an additional helical pitch is added to the twist, leading to an eective stretching of the hydrogen bonds. The colors of the peptide are chosen to highlight the amphiphilicity of the molecule, where red indicates the hydrophobic nature of the alanine amino acids and the charged ends are colored blue.

Figure 6: A schematic image showing the geometry of an An K peptide ribbon consisting of 11 laminated β -sheets. In Figure 6 all the peptides are represented as innitely sti rods. This, in turn, leads to a rectangular cross-section of the formed ribbons. Although the morphology of the peptide bonds induce stiness, the ability of the cα atom of each amino acid to rotate induces peptide exibility. 25 It is therefore safe to assume that such high energy structures, as the right angles in the 3-dimensional model, do not exist in solution. The exible peptides at the edges can relax the structure, assuming a more elliptical cross-section, which justies the chosen model 11


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for SAXS analysis.

Ribbon free energies The driving force of the peptide aggregation is the hydrophobic interaction, associated with the long alanine chains. Therefore, the β -sheets are hydrophobic and spontaneously laminate to minimize the contact with water. For this reason, it is very striking that the number

N ≈ 15 of laminated β -sheets is constant. From the hydrophobic interaction alone, one would expect the preferred N to be innite. Apparently, there is an additional contribution to the free energy that gives a signicant penalty for higher values of N . A description of this thermodynamically controlled width of self-assembled β -sheet ribbons has been presented by Nyrkova et al. 20,21 In adapting this model to the A8,10 K system, we have identied and quantied what we consider the two major free energy contributions to the total ribbon free energy based on what is known about the A8,10 K peptide monomers itself and on the geometrical structure of the ribbons as determined by experiments and shown in Figure 6. The two contributions taken into account are the interfacial free energy of the β -sheets,

surf ace , and a free energy penalty as a result of β -sheet deformation upon lamination, h−bond . The total free energy, tot , is regarded as the sum of these free energy contributions

tot = surf ace + h−bond

(5)

The rst considered contribution is the reduction of hydrophobic surface area in contact with the solution through β -sheet lamination. By approximating a single β -sheet to a twosided hydrophobic sheet with the total area per peptide, 2A = 2lp dp−p , where lp is the length of the stretched peptide monomer, and dp−p is the distance between adjacent peptides within a β -sheet. Unlike in the Nyrkova model where the β -sheets consist of two non-identical sides, we assume here that both sides of the β -sheets are identical in terms of hydrophobicity, due to the fact that all amino acids except one, is the same. From this it becomes clear that

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every further addition of a β -sheet to the structure will bury 2A of the total available surface area, 2AN . Accordingly, the solvent facing surface area scales as 1/N of the total β -sheet surface area. Further assuming that the interfacial tension, γ , of an An K peptide can be approximated to that of a hydrocarbon of length n, this free energy contribution, surf ace , can be calculated through

surf ace =

γ2lp dp−p (0) N

(6)

In the case of A10 K, lp = 3.9 nm and dp−p = 0.45 nm and an interfacial tension of γ = 2.4 ×

10−20 Jnm−2 was used. 28 The calculated free energy is, when using equation 6, normalized per peptide. In the calculations of surf ace we do not consider the surface created at the top and the bottom of the ribbon. As these surfaces are charged their total contribution to the free energy is considered to be of less signicance than the hydrophobic surface. Neither do we consider surfaces at ribbon ends as these only constitute a very small part of the overall ribbon surface. In contrast to the free energy optimization of β -sheet lamination, the free energy penalty is associated with the β -sheet deformation upon lamination. This phenomena can be understood using the previously presented three-dimensional model of the twisted ribbons, Figure 6. As previously described, the central β -sheet, m = 0, follows the propagation vector of the ribbon. However, whereas the midpoints of extended peptides in this central sheet form a straight line, the midpoints of the peptides in β -sheets at m 6= 0 will show a helical contour line. This contour line length, lc , per pitch is determined by lc2 = (2πdβ |m|)2 + λ2 , at the radius, r = dβ |m|, and with the pitch, λ, as extracted from the TEM analysis. The contour line length increases with m. As the peptides have been shown to pack on two-dimensional crystal lattice, we assume that the number of peptides in a β -sheet of a given length is constant throughout the ribbon. Instead the elongation of the β -sheet is proposed to results in a stretch of the hydrogen 13


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bonds holding them together. This stretch, for a specic β -sheet m 6= 0, can be determined as follows:

dp−p (m) = dp−p (0)

lc (m) λ

(7)

where the ratio between the helical contour length of a β -sheet m 6= 0 and λ, is multiplied with the peptide-peptide distance within the central β -sheet, dp−p (0). Here we note that this calculated stretch will be somewhat overestimated as it follows a helical contour length, whereas the shortest peptide-peptide distance would fall on a straight line. However, this overestimation is determined to be smaller than 1 %, and is therefore not further considered in the model. To quantify the free energy penalty associated with the calculated bond stretch, we have chosen a simple model potential, explicitly describing hydrogen bonds between peptide backbones, as described by Irbäck et al., 29

h−bond

" = hb 5

σhb dp−p (m)

12

−6

σhb dp−p (m)

10 #

(8)

With this potential we can calculate the free energy associated with a specic bond stretch

dp−p (m), by assuming an optimal hydrogen bond distance, σhb = dp−p (0). The potential depth is set to hb = 16.71 kJmol−1 . 29 Is association with the calculations of h−bond , we make a couple of assumptions to keep the model simple. First of all, the choice of the optimal hydrogen bond distance is limited in the model to the available information, that is the distance between adjacent peptides in a β -sheet as determined by WAXS. Knowing that the typical hydrogen bond length is somewhat shorter, we are aware that this assumption will lead to an underestimation of the relative stretch of the hydrogen bond. This error will however be small for low values of m. A second assumption regards the directionality of the hydrogen bonds. As these bonds are based in the interactions of two dipoles, they are considered rather directional. The importance of this directionality has specically been shown for estimations of the hydrogen 14


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bond force eld. 30,31 In our model we choose to not consider this angular contribution to the full potential, as the β -sheets are organized in an antiparallel conguration. Again, this assumption is considered valid for β -sheets with very little stretch. Deviations are expected to occur at β -sheets showing a greater deformation. Using the notation and the implied symmetry from Figure 6, we can insert the two contributions into Equation 5 to calculate the total free energy for the ribbon. As previously mentioned, surf ace is already presented per peptide, whereas h−bond is rst multiplied by the total number of hydrogen bonds per peptide, (n + 1), equal to 1 hydrogen bond per amino acid, and further averaged by the total number of β -sheets in the structure,

(n + 1) tot (N ) = surf ace (N ) +

M P m=−M

N

h−bond (m) (9)

to give a total free energy per peptide. tot as a function of the number of laminated β -sheets in the A10 K ribbons is presented in Figure 7, together with the two separate free energy contributions, surf ace and h−bond . A free energy minimum in tot for a nite total number of laminated sheets in the ribbon is observed, indicating that our very simple model seems to capture the major free energy contributions to the total free energy of the twisted ribbons in the A8,10 K system. The free energy minimum calculated using the model occurs at a total of N = 3 β -sheets in the ribbon, corresponding to a nal ribbon width of around 2 nm. Compared to the experimental value of 8 nm, the model seems to underestimate the ribbon width roughly by a factor of 4. This deviation is considered to be a consequence of the very simple model. We note that, while the calculations are only performed for the A10 K peptide, both free energy contributions are proportional to the peptide length. Therefore the position of the tot minimum would be the same for A8 K.

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Figure 7: Total free energy per A10 K petide in a twisted ribbon as a function of N = (2M + 1) laminated β -sheets. The solid line shows tot as the summation of hydrophobic surface minimisation, surf ace , and the free energy increase from the stretching of interpeptide hydrogen bonds, h−bond . This simple model shows a free energy minimum for a certain number of laminated β -sheets corresponding to a nite ribbon width. As seen in Figure 7, the surf ace -term only seems to be signicant for the rst laminated sheets, before it rapidly approaches 0. This implies that the position of the minimum is mainly governed by the h−bond -term. Whereas a couple of assumptions have already been commented on, the main contribution to the results seems to be that of the pitch. As previously mentioned, this experimentally acquired value of the pitch could be aected by the procedure during sample preparation. In comparison to other twisting ribbon systems, the pitch observed for A8,10 K seems to be substantially shorter. 21,26 For the model to predict the experimentally determined width of N ≈ 15, a pitch of roughly λ ≈ 60 nm is needed.

Conclusions The synthetic peptide An K, where n = 8, 10, self-assembles through the lamination of β sheets into highly ordered 1-dimensional ribbon structures, exhibiting a nite and strikingly constant ribbon width of roughly 8 nm. Extending the already existing knowledge of the 16


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ribbons we have been able to show the existence of a ribbon twist with a pitch λ ≈ 15 nm using high resolution TEM. The twist is expected for structures consisting of inherently chiral molecules such as peptides. Adapting a general model regarding the constant width β -sheet containing peptide aggregates to t to the An K system, we have been able to show a clear free energy minimum for a specic number of β -sheets within the structure. Following the work of Nyrkova et al., 20,21 we have analyzed the ribbon free energy quantitatively by considering a combination of interfacial free energy and an elastic stretching deformation arising from the ribbon twist. Using a typical water-oil interfacial tension, a previously derived hydrogen bond potential and an experimentally observed ribbon twist, a reasonable agreement is obtained to the experimental results.

Acknowledgement The authors thank Anna Carnerup at the national Center for High Resolution Electron Microscopy in Lund and Jennifer Sloppy at the Keck electron microscopy facility for the supervision and the expertise during TEM sample preparation and measurements. We further acknowledge UD COBRE NIH-COBRE 1P30 GM110758 for partial support of the Keck electron microscopy facility at the University of Delaware. This research is funded by the

Knut and Alice Wallenberg Foundation, grant number KAW 2014.0052.

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Twisted Ribbon Aggregates in a Model Peptide System

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