Aggregate Size Dependence of Amyloid Adsorption onto Charged Interfaces.

Amyloid aggregates are associated with a range of human neurodegenerative disorders, and it has been shown that neurotoxicity is dependent on aggregate size. Combining molecular simulation with analytical theory, a predictive model is proposed for the adsorption of amyloid aggregates onto oppositely charged surfaces, where the interaction is governed by an interplay between electrostatic attraction and entropic repulsion. Predictions are experimentally validated against quartz crystal microbalance-dissipation experiments of amyloid beta peptides and fragmented fibrils in the presence of a supported lipid bilayer. Assuming amyloids as rigid, elongated particles, we observe nonmonotonic trends for the extent of adsorption with respect to aggregate size and preferential adsorption of smaller aggregates over larger ones. Our findings describe a general phenomenon with implications for stiff polyions and rodlike particles that are electrostatically attracted to a surface.

Introduction

A large number of peptides and proteins self-assemble into elongated and highly ordered structures, rich in β-sheets, which are generally termed amyloid fibrils.[1−3] The formation process is controlled by amino acid sequence, solution conditions such as pH, and concentration of salts or cosolutes as well as by the presence of surfaces in contact with the solution.

In biology, protein aggregation occurs in environments containing proteins and protein complexes, glycoproteins, nucleic acids, ribosome particles, and lipid membranes. The protein self-assembly process and the final composition of the amyloid aggregates are affected by this environment.[4−7] Likewise, protein aggregation influences properties of existing self-assembled entities, for example, the structure and integrity of biological membranes.[4,8,9]

Amyloid formation is associated with many human diseases.[2] To understand cellular toxicity, it is important to resolve the role played by the interactions between proteins, in different aggregation states, and other components of the complex environment. Several amyloid proteins, e.g., Aβ, α-synuclein, and IAPP, are surface active and adsorb to solid surfaces, to air–liquid interfaces, and to lipid bilayers.[10−14] Adsorption of protein aggregates to a membrane interface is the first step toward cell permeabilization, and in this work we investigate the interaction between elongated aggregates of charged peptides and an oppositely charged surface.

Aggregate size is one possible determinant of the observed cytotoxicity. Several studies have shown that oligomers and short, fragmented fibrillar aggregates are more potent in permeabilizing cell membranes, and in reducing cell viability, compared to longer fibrils.[15−18] Fragmentation of β2-microglobulin, α-synuclein, and lysozyme fibrils into shorter aggregates has been shown to cause increased cellular damage,[19] and the enhanced cytotoxicity has been related to the enrichment in fibril extremities upon fragmentation.[19,20] Indeed, interactions between the extremities of short β2-microglobulin fibrils and negatively charged liposomes have been reported to induce membrane distortions via lipid extraction from the bilayer.[20] Analogously, short α-synuclein fibrils have been shown to associate with negatively charged lipid bilayers and form protein–lipid coaggregates.[4] Moreover, it has been reported that long (100–400 nm) and short (10–100 nm) α-synuclein aggregates have comparable affinity to the plasma membrane of mammalian cells, and they bind with predominantly parallel orientation to liposomes composed of brain lipids.[21] Lateral binding has also been observed for huntingtin exon 1 fibrils of length 40–120 nm, while shorter fragments displayed low affinity to both liposomes and cells.[21]

We use the quartz crystal microbalance–dissipation (QCM-D)[22] technique to measure the interaction of Aβ1–40 in either monomeric or fibrillar form with fluid lipid bilayers. Our model system consists of a combination of Aβ1–40 and a positively charged POPC:DOTAP 3:1 lipid bilayer. Experimental results are interpreted using molecular simulations and an analytical theory, based on the representation of aggregates as charged line segments. Molecular simulations show how the adsorption is influenced by aggregate size, and decay length of the electrostatic attraction to the surface, which is controlled by the bulk ionic strength. The analytical line segment theory captures the dependence of surface–aggregate interaction on bulk ionic strength and aggregate size. Using generalized van der Waals theory, we calculate the surface excess of line segments, a quantity which can be directly related to the QCM-D data.

Materials and Methods

1-Palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) and 1,2-dioleoyl-3-trimethylammonium–propane (DOTAP) were purchased lyophilized from Avanti Polar Lipids Inc. (Alabaster, AL). All chemicals were of analytical grade.

Amyloyd-β Peptides and Fibrils

The amyloid-β peptide Aβ1–40 was expressed in Escherichia coli and purified according to Walsh et al.[23] Aβ1–40 monomers were isolated by size exclusion chromatography in 20 mM sodium phosphate buffer, 0.02% NaN3, 0.2 mM EDTA, pH 7.4. For samples with salt concentration 150 mM, NaCl was added to the monomer solution on ice from a 30× concentrated stock. Fibrils were prepared by keeping the monomer solution at 37 °C with moderate shaking (IKA-VIBAX-VXR motor operating at 200 rpm) in low-binding tubes (Axygen). Fibrils were sonicated for 30 min with a tip sonicator using cycles of 1 s pulses and 1 s waiting time. The length of sonicated fibrils was characterized using atomic force microscopy (AFM) (Supporting Information, Figure S4). The peptide concentration of the samples used for QCM-D experiments was 4 μM and for AFM 10 μM. The contribution of the sodium phosphate buffer to the ionic strength of the solutions is 50 mM.

Bilayer Formation

For the positively charged lipid bilayer, a POPC:DOTAP 3:1 mixture was prepared in chloroform:methanol 2:1 v:v and deposited as a thin film on glass under flow of N2 gas and dried under vacuum overnight. The lipid film was dispersed in 200 mM NaCl aqueous solution to obtain a lipid concentration of 0.5 mg/mL. For the neutral lipid bilayer, POPC was dispersed in water to obtain a lipid concentration of 0.5 mg/mL. Lipid dispersions were probe-sonicated in an ice bath for 10 min alternating 5 s pulses with 5 s of cooling. The clear vesicle dispersions were centrifuged for 20 min at 2000 rcf to remove debris from the probe tip. The supernatant which contains small unilamellar vesicles was used for the deposition of the lipid bilayer.

QCM-D Experiments

The Q-sense E4 from Qsense (Göteborg, Sweden) is the QCM-D instrument used for the experiments. Measurements were performed in parallel in four cells thermostated at 25 °C with quartz crystals covered by a thin gold surface coated with 50 nm SiO2 (QSX 303, Q-sense). The crystals were stored in 2% w/v SDS solution for at least 1 h, rinsed in Milli-Q water followed by ethanol, and dried with N2 gas. Finally the crystals were treated in the low pressure (0.02 mbar) chamber of a plasma cleaner (Harrik Scientific Corp., Pleasentville, NY, model PDC-3XG) for 10 min. The crystals were mounted in the cells and equilibrated with water until steady frequency and dissipation responses were observed. In the experiments with the positively charged lipid bilayer, an injection of a 200 mM NaCl aqueous solution, with flow rate of 300 μL/min, preceded the injection of the vesicle dispersion, with flow rate of 100 μL/min. The high ionic strength of the dispersion facilitates vesicle formation and the deposition of the lipid bilayer by screening the electrostatic repulsions between the positively charged DOTAP head groups. The supported lipid bilayer was equilibrated in the buffer solution, and finally the freshly isolated Aβ1–40 monomers or the sonicated Aβ1–40 fibrils were injected. Both buffer and protein solution flowed through the cells at a rate of 50 μL/min.[24−26]

MC Simulations

We performed Metropolis Monte Carlo (MC) simulations using the Faunus framework.[27] A single aggregate (Figure ) is placed in a rectangular box of volume 200 × 200 × 100 nm3, and a wall of uniform charge density is located at z = 0 nm. The potential energy contribution of the ith bead of radius R at a distance r from the surface is given by the linearized Gouy–Chapman potential[28]where z is the partial charge number of the amino acid, e is the elementary charge, and λD = 3.04/√cs Å is the Debye screening length, at T = 298 K, for the surrounding monovalent (1:1) salt of molar concentration, cs. The surface potential is βϕe = 2 asinh(2πσλBλD/e), where λB = 7.1 Å is the Bjerrum length of water and β = 1/kBT is the inverse thermal energy, while σ = 1/266 e Å–2 is the surface charge density of a 3:1 POPC:DOTAP lipid bilayer.[29] The internal degrees of freedom of the aggregates are frozen, and the configurational space is sampled by attempting roto-translational moves of the rigid body. Sampling is enhanced using the Wang–Landau method,[30] where the reaction coordinate is the separation of the center-of-mass of the aggregate from the surface, r.

Figure 1

Cross sections and side views of coarse-grained aggregates of N Aβ1–42, Aβ17–42, and Aβ1–40 peptides. Each bead corresponds to an amino acid that can be neutral (white), cationic (blue), or anionic (red). Fibrils of the Aβ17–42 fragment present a smooth rodlike structure. The additional 16 N-terminal residues of the Aβ1–42 peptide are mostly charged or titratable, and they decorate the core of Aβ1–42 fibrils with flexible chains which extend in the surrounding solution,[32,33] resembling a polyelectrolyte brush. Aβ17–42 and Aβ1–42 structures have 2-fold symmetry, whereas the fibril made of Aβ1–40 peptides has three peptides per cross section and 3-fold symmetry.

To study the effect of counterion condensation, a single coarse-grained aggregate is placed in a cylindrical box with the fibril principal axis aligned to the cylinder axis. We performed MC simulations of the static aggregate surrounded by explicit mobile monovalent (cs = 0.25, 0.5, and 5 mM) or divalent (cs = 2 mM) counterions of radius 1.9 Å.[31] We calculated the average total charge within coaxial cylinders of increasing radii enclosing the aggregate and the ions. The effective charge of the aggregate is determined as the total charge at the radial distance corresponding to the peak in the counterion density profile.

Results and Discussion

QCM-D Experiments

QCM-D provides measurements of the temporal variation in frequency (ΔF) and energy dissipation (ΔD) of a quartz crystal resonator when the lipid or protein and associated solvent molecules adsorb on the sensor surface. Figure shows the measured |ΔF|/n and ΔD for solutions of Aβ1–40 monomers and fragmented fibrils in the presence of a supported POPC or POPC:DOTAP 3:1 lipid bilayer. Signals are recorded for various overtone numbers, n, of the fundamental frequency of the quartz crystal. The zero value in Figure corresponds to the stationary response after deposition of the lipid bilayer and equilibration with the buffer solution. Figure A,B shows that negatively charged Aβ1–40 monomers and fragmented fibrils in solutions of pH 7.4 and 50 mM ionic strength, cs, do not adsorb to neutral POPC lipid bilayers. Therefore, despite being surface active, the association of Aβ to fluid lipid bilayers is not controlled by hydrophobic interactions.[34] Panels d, e, and f of Figure show that in the presence of the positively charged POPC:DOTAP 3:1 lipid bilayer no significant changes in |ΔF|/n and ΔD are detected upon addition of Aβ40 monomers, neither at high nor at low ionic strength. Similarly, when sonicated fibrils are added at high salt condition, no adsorption on the deposited bilayer is detected. However, adsorption clearly occurs after the injection of sonicated Aβ1–40 fibrils in the buffer with no added NaCl, as directly inferred from the decrease in ΔF/n by around 5–10 Hz and from the simultaneous increase in ΔD in Figure C. The dissipation shift and the n-dependent response observed upon injection of Aβ1–40 fibrils at low ionic strength conditions indicate that the adsorbed fibril layer on the positively charged lipid bilayer is viscoelastic and acoustically coupled to the aqueous solution. Hence, the experimental data were analyzed with a one-layer extended viscoelastic Voigt-based model[35,36] (Supporting Information, QCM-D Data Analysis). In the continuum model, the adsorbed fibrils are represented by a viscoelastic layer, while the supported lipid bilayer is considered to be unaffected by fibril adsorption. The fibril layer is in contact with a semi-infinite solution modeled as a Newtonian fluid. The extended model accounts for a linear frequency dependence of the viscosity and shear modulus of the fibril layer,[37] expressed in terms of frequency factors. The parameters evaluated in the fitting procedure are the frequency factors and, more importantly, the time evolutions of the shear modulus, viscosity, and mass per area of the fibril layer (Supporting Information, Table S1 and Figure S3). The fitted quantities significantly increase upon injection of Aβ1–40 fibrils, and the equilibrium wet mass of adsorbed hydrated fibrils is estimated to be of 660 ± 30 ng cm–2.

Figure 2

Frequency (blue lines) and dissipation changes (red lines) for overtone numbers 5, 7, and 9 after injection of (A, C, E) fibrils solutions and (B, D, F) monomer solutions of peptide concentration 4 μM. The zero values of ΔF/n and ΔD correspond to the supported lipid bilayer equilibrated with the buffer solution. No adsorption is observed for monomers in both low (cs = 0.05 M) and high salt conditions (cs = 0.2 M) on POPC as well as on POPC:DOTAP 3:1 bilayers. For fibrils, significant adsorption is detected only at cs = 0.05 M on the POPC:DOTAP 3:1 bilayer (C). The corresponding ΔF/n and ΔD are fitted to the one-layer extended Voigt-based model (C, black lines). The yellow symbols facilitate the comparison with the predictions of the line segment model shown in Figure B.

Figure 6

(A) Surface excess of aggregates, Γ, and (B) adsorbed amount, Δm, as a function of salt concentration, cs, and aggregate size, N. Values are calculated using the line segment model for Aβ1–40 aggregates (zm* = −1.35, lm = 2.2 Å, s0 = 19.3 Å, s1 = 72.7 Å, and ν = 1556). The bulk peptide number density is ρ = 2.4 × 10–6 nm–3. The yellow symbols in (B) highlight the conditions of N and cs explored in the QCM-D experiments of Figure . Contour lines connect conditions of fibril length and ionic strength yielding same values of Γ or Δm.

Molecular Simulations

Computer simulations are used to calculate the interaction free energy between Aβ elongated aggregates and a planar surface as a function of aggregate size and salt concentration. The strategy is to first use all-atom molecular dynamics (MD) simulations to relax the structure of fibrillar assemblies (Supporting Information), which are then coarse-grained to the amino acid level and used in Metropolis MC simulations of surface–aggregate interactions.

Previous studies provided evidence of a difference in morphology between amyloid oligomers, protofibrils, and fibrils.[38−40] To take polymorphism of amyloid aggregates into account, our in silico investigation extends over three fibril architectures (Figure ): two structures with 2-fold symmetry for Aβ1–42 peptides[33] and Aβ17–42 fragments[41,42] as well as a structure with 3-fold symmetry for Aβ1–40 peptides, which is predominant in Alzheimer’s disease human brain tissues.[43,44] Assemblies of around 200 peptides are generated by stacking fibril segments obtained from the protein data bank (PDB entries: 5KK3, 2M4J, and 2BEG, after Zheng et al.[42]), and simulated annealing[45] is used to minimize the energy of the large structures. At pH 7.4, Aβ1–42 and Aβ1–40 have the same net charge number, zm, of −3, while the Aβ17–42 fragment has zm = −1. All-atom elongated aggregates are coarse-grained to rigid models where each amino acid is represented by a bead that can be neutral, cationic, or anionic. Approximating aggregates as rigid bodies is justified by the stiffness of Aβ fibrils, which have persistence lengths of micrometers.[43,46,47] While the distinction between oligomer and fibril may be operational, structural, or size-based,[48] for our simulations we generate aggregates of various numbers of monomers, N, by cross-sectional slicing of the long fibril. This approach is analogous to the physical fibril fragmentation process that has previously been used to show the enhanced cytotoxic potential of small-sized amyloid fragments.[19]

In MC simulations of the rigid aggregates in the presence of a charged, planar interface, we evaluate the surface–aggregate interaction free energy usingwhere the angular brackets denote a canonical ensemble average over the orientational degrees of freedom of the aggregate, Ω, u is the interaction of the ith amino acid with the surface at a distance r, see eq , and r is the aggregate mass center to surface separation.

The relative protein concentration in the interfacial volume, with respect to the bulk, is quantified by the surface excess, Γ.[28] An aggregate surplus (Γ > 0) or depletion (Γ < 0) is determined by surface–protein and protein–protein interactions. For a weakly interacting system, Γ can be described by Henry’s law: Γ ≈ KHAρ/N, where ρ is the bulk peptide number density and KHA is Henry’s law constant, related to w(r) through the Mayer integralFigure A shows free energy profiles obtained from eq for Aβ1–42 assemblies of various N at cs = 0.4 M. For smaller aggregates, the interaction is short-ranged and repulsive. Conversely, free energy profiles for larger aggregates are characterized by a minimum for the fibril in proximity of the surface and by a free energy barrier between surface and bulk solution. These profiles result from the interplay between surface–aggregate electrostatic attraction and the entropic repulsion due to the decrease in rotational degrees of freedom as the aggregate approaches the surface. Both attractive and repulsive forces are heightened with increasing fibril length. The free energy barrier increases with N, suggesting that in real systems larger aggregates might be kinetically trapped in solution. The subtle kinks occurring at r around 5, 9, and 18 nm for aggregates of 32, 64, and 128 peptides, respectively, reflect the possibility for the aggregate to be oriented perpendicularly to the surface. This orientation is favored by end-point electrostatic attraction (Figure S7) as well as by the absence of steric hindrance from the surface. At cs = 0.4 M, aggregates of N = 64 preferentially bind to the surface through their extremities, while they bind laterally at cs = 0.35 M. Aggregates of N = 128 bind laterally at all explored cs values. This illustrates that cs and N modulate the preferential orientation of the adsorbed aggregate.

Figure 3

(A) Angularly averaged interaction free energy, w(r), as a function of surface–aggregate separation for Aβ1–42 assemblies of various size, N, and 0.4 M salt concentration, cs. (B) Henry’s law constants, KHA, calculated from MC simulations (circles) and from the line segment model (lines) for aggregates of Aβ1–42 of increasing N and at different cs. Data points labeled n.s. are from calculations with a neutral surface.

Figure B shows Henry’s law constants, KHA, calculated from MC simulations for various N and cs. At conditions where the electrostatic interactions are negligible, i.e., at high cs and for neutral surfaces (n.s.), the surface–aggregate interaction is repulsive at all separations and more so for longer fibrils. This is expected as the interaction is controlled solely by the orientational entropy loss, when fibrils approach the interface. At low cs, the trend reverses, and long fibrils adsorb to the interface with interaction free energies of several kBT. Here, the electrostatic attraction dominates the interaction, greatly exceeding the entropic cost of aligning the rodlike fibrils parallel to the surface. For intermediate cs, the interplay between entropic loss and electrostatic attraction results in a more complex behavior: KHA varies nonmonotonically with N (brown line in Figure B), indicating that fibrillar assemblies of certain lengths may be repelled while others attracted. The same conclusion is drawn by inspecting the free energy profiles in Figure A.

We now compare KHA values for aggregates of the Aβ1–42 fragment with the corresponding data for rigid aggregates of Aβ17–42 and Aβ1–40 peptides. The three amyloids have architectures differing in cross-sectional area and symmetry, line charge density, and extent of exposure of charged residues to the surface. Nonetheless, Figures and 4 show that aggregates of Aβ1–42, Aβ1–40, and Aβ17–42 display similarities in the dependence of surface interaction on cs and N. Notably, we observe fair agreement between the dependence of KHA on N and cs for Aβ1–42 and Aβ1–40 aggregates which have different symmetry but similar line charge densities (Figures B and 4A). Therefore, it seems reasonable to approximate the elongated rigid aggregates by line segments with a characteristic line charge density. We begin by studying the adsorption of a negatively charged aggregate on a positively charged surface, to conclude that the underlying interactions are independent of the sign of the net charge of the interacting entities, as long as they are oppositely charged. As a consequence, the line segment model developed below is equally applicable to the case of a positively charged fibril interacting with a cell membrane.

Figure 4

Henry’s law constants, KHA, calculated from MC simulations for (A) Aβ1–40 and (B) Aβ17–42 aggregates of increasing N and at different cs. Data points labeled n.s. are from calculations with a neutral surface. Lines are calculated using the line segment model.

Line Segment Model

In the following, we describe how the conceptual line segment model for surface–aggregate interactions is constructed. Consider a freely rotating line segment of length L, with the center point located at a distance r away from a planar surface. The entropy change corresponding to the reduced number of available rotational states is related to the relative area of the spherical belt spanned by the ends of the line segment with respect to a sphere of diameter L.Using the expression for the electrostatic interaction between a line charge and the unperturbed double layer of a charged surface,[49] the total surface–line interaction free energy is approximated bywhere s is the distance of closest approach between aggregate and surface, while z is the net charge number of the aggregate. Each monomer contributes to the aggregate length and charge number by lm = L/N and zm = z/N, respectively. The r–9 term introduces a soft repulsion between rod and surface.[50] To allow for the difference in dimensionality between oligomers and longer fibrils, s is modeled by a smooth function varying between s0 and s1, s = s0 + (s1 – s0) tanh(N/ν). s0, s1, and ν are determined from global least-squares fits to the simulated KHA values for high cs (Table S2) while lm values of 2.8, 2.2, and 2.6 Å are obtained from long Aβ1–42, Aβ1–40, and Aβ17–42 coarse-grained aggregates, respectively. Figures B and 4 show KHA as a function of N as obtained from eqs and 5 for the three different amyloids. The line segment model closely reproduces the MC simulation results: adsorption is enhanced by increasing N and decreases with increasing cs. Further, the model captures the oscillating trends at cs values where entropic repulsion and electrostatic attraction are of comparable magnitude. The agreement between molecular simulation results and the analytical model indicates that the main features of the adsorption behavior are independent of molecular-level structural details, such as the discrete charge distribution on the fibril surface. Hence, the line segment model is also valid for positively charged elongated aggregates adsorbing onto a negatively charged surface.

The calculation of Γ is based on the description of the adsorbed particles as a two-dimensional fluid, using a generalized van der Waals approach. The excluded area per monomer is set to 3σ–1, implying that the maximum coverage occurs when the surface charge is neutralized by the adsorbed particles. Adsorbed fibrils laterally repel each other as cylinders of line charge density z/L and diameter s(51)where K0 is the zero-order modified Bessel function of the second kind. Γ is given by the implicit equation[52]where KHA was defined in eq , while â is the mean-field energy constant for the adsorbed aggregates â = −1/2∫∞dr 2πrβw(r) surface. The charge density of rodlike polyelectrolytes in solution is compensated by counterion condensation,[53,54] and highly charged rods are likely to release only a small fraction of condensed ions upon adsorption onto a weakly charged surface of opposite sign.[55] Therefore, for the calculation of Γ, we consider an effective charge number per monomer, zm*, estimated from Monte Carlo simulations of a single fibril surrounded by its counterions. Figure shows the cumulative sum of aggregate and counterion charges as a function of the radial distance from the aggregate axis, Rc. Even with monovalent counterions, the effective charge of Aβ1–42 and Aβ1–40 aggregates is significantly reduced by counterion condensation. In contrast, for the less densely charged Aβ17–42 aggregate, counterion condensation occurs only with divalent ions.

Figure 5

Total charge of aggregate plus counterions (red lines) within coaxial cylinders as a function of the cylinder radius, Rc. Gray and blue lines are the total charge of the aggregate and counterions, respectively. Counterions are monovalent at cs = 0.25 mM (solid lines) or divalent at cs = 2 mM (dashed lines). Aggregates consist of (A) N = 120 Aβ1–42 peptides, (B) N = 150 Aβ1–40 peptides, and (C) N = 130 Aβ17–42 peptides. The dashed vertical lines indicate the positions of the peaks in the counterion radial density profiles which are used to determine the effective charge of the aggregates.

For a wide range of N and cs, Figure A displays Γ values predicted by the line segment model using eq and the parameters derived for Aβ1–40 aggregates (Table S2). Solid contour lines connect conditions of N and cs yielding constant Γ values. The line of Γ = −0.003 nm–2 indicates that for high cs the surface excess of aggregates varies nonmonotonically with N. Points on the left-hand side of the line of zero-Γ correspond to conditions where the aggregates are attracted to the surface. The contour lines of Γ = 0.5, 1.0, and 1.5 nm–2 highlight that our model predicts larger Γ values for oligomers than for fibrils, at low-to-intermediate cs. This stems in part from the decrease, with increasing N, of the number of aggregates required to neutralize the surface.

Figure B displays line segment model estimates of the adsorbed amount, Δm = NΓMW/NA, where MW is the molecular weight of the Aβ1–40 peptide. This conversion allows for direct comparison with QCM-D experimental results. Contrary to what we observe for Γ, at low cs, Δm is higher for long fibrils than for oligomers. However, Figure shows that at physiological ionic strength both Γ and Δm are the highest for a range of relatively short aggregates with maximum values for lengths of 20 and 70 nm, respectively.

Figure 7

(A) Surface excess of aggregates, Γ, and (B) adsorbed amount, Δm, calculated as a function of aggregate length, L, from the line segment model for Aβ1–40 aggregates (zm* = −1.35, lm = 2.2 Å, s0 = 19.3 Å, s1 = 72.7 Å, and ν = 1556) and four values of cs. The surface charge density is σ = 1/266 e Å–2, while the bulk peptide number density is ρ = 2.4 × 10–6 nm–3.

For crystals with fundamental frequency of 5 MHz, the Sauerbrey equation (Supporting Information, eq S1) gives an estimate of around 10 ng cm–2 for the smallest adsorbed wet mass yielding a ΔF/n signal significantly lower than zero. Sonicated fibril samples at low and high cs have length distributions between 40 and 130 nm (Figure S4), corresponding to N between 180 and 590. Figure B shows that Δm is large in a range of cs values which extends to cs = 0.17 for N = 250. Beyond the upper bound of this cs interval, the screened surface–aggregate electrostatic force and the entropic repulsion counterbalance. As indicated by the yellow symbols in Figures and 6B, the line segment model reproduces the QCM-D results, predicting significant adsorption for aggregates at low ionic strengths, while the binding affinity is considerably lower for monomers and for sonicated fibrils at cs ≥ 0.2 M.

At cs = 0.05 M, the predicted adsorbed dry peptide mass is Δm ≈ 90 ng cm–2, i.e., 13.6% of the wet mass determined from QCM-D experiments. Assuming this composition, the protein layer has density of 1.05 g cm–3 and a thickness of 6.3 ± 0.3 nm, implying that, on average, the fibril principal axis forms angles smaller than 10° with the surface. The QCM-D signals displayed in Figure are characteristic of a highly hydrated protein layer, and the quasi-parallel fibril orientation is expected for a strong surface–aggregate attraction.

Summary and Conclusions

In summary, the proposed model offers a description of the adsorption of rigid-rod-like amyloid aggregates to oppositely charged surfaces and predicts that small aggregates can lead to higher surface excess and adsorbed amount than long fibrils at physiological pH and ionic strength. This theoretical result offers a feasible explanation for why longer fibrils might be less cytotoxic than shorter ones,[16,17] albeit oligomer flexibility and membrane structure may also play significant roles.[56]

The model predicts that at low ionic strength, cs, the number of adsorbed molecules at the surface is larger for short than for long aggregates, while the dependence of the extent of adsorption on N is less pronounced at higher cs. Moreover, large N and low cs favor lateral surface binding while end-point binding is favored by shorter length and higher cs.

We show that small changes in solution ionic strength as well as fibril line charge density and length have a large impact on amyloid adsorption. This marked sensitivity on system conditions contributes to explain the seemingly contradictory experimental evidence regarding the length dependence of the affinity and preferential orientation of positively charged amyloid aggregates to plasma membranes.[19−21] In particular, amyloid adsorption is likely to be affected by the low concentrations of divalent cations present in the extracellular space in vivo as well as in cell culture media. Compared to monovalent ions, divalent counterions reduce to a larger extent the effective line charge density of the aggregate[54] (Figure ). However, they also mediate the interaction between adsorbed like-charged aggregates and may lead to a net lateral attraction at the interface.[57]

Additionally, the presented adsorption mechanisms are of interest for colloids and molecules that can be likened to rigid charged elongated particles—DNA strands and polyelectrolytes,[58−61] rodlike particles,[62−64] and amyloid aggregates in adhesive biofilms, spider silk, aggregation of milk proteins, and new functional materials.[65−67] For systems where such particles interact with an oppositely charged surface, the presented results provide insight into the complex dependence of the adsorbed amount on particle size and solution ionic strength.