Probing the Interactions of Intrinsically Disordered Proteins Using

Mar 30, 2015 - proteins lack a fix three-dimensional structure but can interact with multiple partners .... aggregation at up to 2 M NaCl and that the...
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Probing the Interactions of Intrinsically Disordered Proteins Using Nanoparticle Tags Stive Pregent,*,†,‡ Amir Lichtenstein,‡ Ram Avinery,† Adi Laser-Azogui,† Fernando Patolsky,‡,§ and Roy Beck*,†,‡ †

School of Physics and Astronomy, ‡Center for Nanoscience and Nanotechnology and §School of Chemistry, Tel Aviv University, Tel Aviv, Israel S Supporting Information *

ABSTRACT: The structural plasticity of intrinsically disordered proteins serves as a rich area for scientific inquiry. Such proteins lack a fix three-dimensional structure but can interact with multiple partners through numerous weak bonds. Nevertheless, this intrinsic plasticity possesses a challenging hurdle in their characterization. We underpin the intermolecular interactions between intrinsically disordered neurofilaments in various hydrated conditions, using grafted gold nanoparticle (NP) tags. Beyond its biological significance, this approach can be applied to modify the surface interaction of NPs for the creation of future tunable “smart” hybrid biomaterials. KEYWORDS: Gold nanoparticles, intrinsically disordered proteins, neurofilaments, small-angle X-ray scattering, spectroscopy

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Inorganic colloidal nanoparticles (NPs) are favored candidates for many applications due to their chemical and physical properties.16−26 However, in solution these NPs are subjected to a range of forces, mainly van der Waals, hydrogen bonding, electrostatic interactions, and repulsive entropic forces. The attractive forces tend to lead to aggregation of the NPs, which degrade their unique properties. In order to counteract these forces and to stabilize NPs in solution, various routes have been studied. For example, synthetic- and biopolymers are commonly grafted onto the NP surfaces to modify interparticle interactions by including additional repulsive forces.16−25 A fundamental hurdle challenging the synthesis of advanced “smart” and responsive colloidal systems is that synthetic polymers are largely based on robust, high-energy bonds that are difficult to chemically modify postsynthesis. In contrast, biopolymers in general, and IDPs in particular, are assembled through the combined effect of many weak interactions with specificity and in many cases reversibility. Moreover, through enzymatic reactions, these interactions can be modified postsynthesis in order to alter their functionality.9 IDPs are natural candidates to act as surface modifiers to colloidal NPs with multiple weak, but tunable, protein−protein interactions. Known and desirable traits of given IDPs can also impart material properties that cannot be achieved with synthetic or polymer materials that tend to bind with nonreversible, high

growing interest in proteomics has been recently directed to proteins that contain large intrinsically disordered domains.1−4 It is now estimated that the human genome encodes 40−50% of intrinsically disordered proteins (IDPs); their functionality derives from their disordered nature.2−6 IDPs interact via a combination of multiple weak bonds, which are sensitive to the environment and are generally reversible. Most IDPs are highly charged and hydrophilic. Oppositely charged residues in IDPs can induce nonspecific ionic bridging and transient attraction.7−9 While neutral and hydrophilic IDPs form collapsed random structures, hydrophobic IDPs can collapse into molten globules characterized by mobile side chains and unstable tertiary structures.1,10,11 In all cases, the flexibility of IDPs enables its interaction with a variety of biological molecules (e.g., proteins, carbohydrates, and nucleic acids). Furthermore, they can promote cooperative binding, act as allosteric activators, or entropic bristles to promote aggregate solubility.1,12 IDPs can also fold upon binding to proteins and surfaces and move through narrow pores and channels. Such interactions between IDPs, and between IDPs and other proteins and biological molecules, depend strongly on their environmental (buffer) conditions. This adds another degree of freedom, or constraint, for tunable interactions.2−4,13−15 The structural plasticity of IDPs enables them to adopt many conformations and multiple weak, nonspecific, and reversible bonds between themselves or with substrates. However, the same biological beneficial structural plasticity restricts the possibly to study them through conventional structural techniques such as protein crystallography. © XXXX American Chemical Society

Received: January 7, 2015 Revised: March 11, 2015

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DOI: 10.1021/acs.nanolett.5b00073 Nano Lett. XXXX, XXX, XXX−XXX

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Figure 1. Schematic representation of the aggregation of NFLt modified GNPs under environmental stimulus such as salt concentration modification.

energy bonds.22 Moreover, NPs bare physical properties can serve as powerful tags to study previously unexplored forces and interactions between IDPs in solution. Neurofilaments (NFs) are the main cytoskeletal constituents of myelinated axons.27,28 They are composed of three molecular weight subunits: NF-L (low), NF-M (medium), and NF-H (high), which self-assemble to form a 10 nm thick filament backbone, with protruding side-arms formed by their C-terminus tails.28−31 These protruding C-terminus tails are intrinsically disordered and highly charged but interact with each other via a combination of electrostatic long-range repulsion, and short-range electrostatic ionic-bridging attraction between oppositely charged residues.7,31−33 The interactions between neighboring NFs dictate the interfilament distance as well as the structural cohesion between filaments and thus have an important biological role.31,34,35 However, the interaction between NFs is far more complicated as the structured filament backbone is by itself flexible and charged. Thus, interactions between the intrinsically disordered tails and the backbone can compete with tail−tail interactions.8,36 In this work, we isolated NF-L tails (NFLt) and grafted them onto gold nanoparticles (GNPs) and surfaces (Figure 1). This approach allows us to understand the role of the intrinsically disordered regions in NFs9 and to gain an in-depth understanding of their function, and dysfunction, as in the case of neurodegenerative diseases.27,30,37−39 The study also allows us to decouple the role of the NF backbone from the disordered tails in various environmental conditions. Our results show that GNPs decorated with NFLt can be stabilized against aggregation at up to 2 M NaCl and that the stabilization mechanism is ion-specific. Mouse NFLt sequences were produced by polymerase chain reaction (PCR) with the full DNA sequence of mouse NF-L (in a pET 30a plasmid) as template. The forward PCR primer (Integrated DNA Technologies, Inc.) included a cysteine and N-Avitag sequence prior to NFLt. The sulfhydryl side chain of the cysteine group can readily serve as a coupling group to gold surfaces and gold NPs. The N-Avitag functional groups may be used for alternative binding to specific streptavidin substrate. After protein expression in E. coli and purification by anion exchange chromatography (see Supporting Information), NFLt fragments were grafted on to GNPs by dialysis in the final assembly buffer. At pH 7, the 162 amino-acids NFLt (18 kDa) has a −37.8e net negative charge. Nevertheless, 18 basic aminoacids within the sequence provide NFLt with its polyampholytic nature.

We used a quartz crystal microbalance (QCM) to quantify the grafting density of the NFLt onto gold surfaces. Then, the interactions between NFLt were monitored by grafting them onto GNPs and following their plasmon resonance change by UV−visible spectroscopy in solution. Attraction of NFLt caused GNPs aggregation; the NP absorption spectra changed from a maximum around 530 nm to over 620 nm. Such changes can also be easily monitored visually, because the solution color changes from pink to blue. In addition, we followed the aggregation processes of the NFLt-modified GNPs by electron microscopy and small-angle X-ray scattering (SAXS). We first studied the binding of NFLt onto gold surfaces, and the conditions needed to attain monolayers of NFLt with adjustable grafting density by QCM (see Supporting Information). When the proteins were bound directly, via thiol bonds, onto the gold surface in 20 mM 2-(Nmorpholino)ethanesulfonic acid (MES), pH 7, we obtained a monolayer grafting density of up to 0.069 ± 0.001 proteins/ nm2 (14.4 nm2/protein). When the gold surface was first modified with a sulfosuccinimidyl 4-[N-maleimidomethyl] cyclohexane-1-carboxylate heterolinker (see Supporting Information), in PBS, pH 7.4, and in the presence of 0.2 w/v % SDS, we reached a higher density of up to 0.15 ± 0.01 proteins/nm2 (6.7 nm2/protein). This suggest that the use of such heterolinker and SDS increases the grafting density of the NFLt on gold surfaces and that brushlike structures can be obtained on the gold surfaces. These results are higher than the reported 0.02 proteins/nm2 (50 nm2/protein) obtained by Srinivasan et al.9 for binding NFH tails, which are longer by about 480 amino acids. Given previous measurements on NFH tails9 and native NFL8 filaments we estimate that NFLt decorate the surface with brush thickness between 5.9 to 15 nm. Following the results from QCM, we grafted NFLt in 20 mM MES buffer at pH 7 to 20 nm diameter GNPs (GNP20, BBI solutions). These GNPs are stabilized by a citrate layer, which gives them a negatively charged surface and prevents aggregation of the particles in low salt concentration buffer. The GNPs were thus separated from their storage solution (water) by centrifugation and redispersion into 20 mM MES pH 7. PBS buffer leads to aggregation due to its high salt concentration and precipitation of bare GNPs. We therefore chose to store GNPs in 20 mM MES buffer at pH 7. The NFLt were added in excess to the GNPs and incubated at room temperature overnight. The NFLt display a cysteine group at their N-terminus and thus readily bind onto the GNPs surface. The GNP concentrations ranged between 0.4 and 1.16 nmol/L B

DOI: 10.1021/acs.nanolett.5b00073 Nano Lett. XXXX, XXX, XXX−XXX

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Nano Letters for all experiments, unless stated differently. As shown later on, grafting of the NFLt dramatically modifies the interparticle interactions under different environmental conditions. Unmodified GNPs are decomposed within few minutes in the presence of excess NaCN. Nevertheless, the modified GNPs-NFLt are stable over few days period with 1 mM NaCN (see Supporting Information). Those results strongly support high grafting density of the NFLt that protect the GNP from decomposition. Such days long shielding is far more robust than the steric stabilization using polyethylene glycol (PEG), protecting GNPs for only few hours.40 Furthermore, this slow decomposition reaction may be useful to selectively remove hybrid particles with lower grafting densities. To demonstrate the influence of grafted NFLt units on the GNP-GNP interactions, we increased the ionic strength of the solution. In the presence of 160 mM NaCl, unmodified GNPs rapidly aggregate due to screening of electrostatic repulsive interactions. The aggregation can be readily detected visually by a color change of the solution from pink to blue (Figure 2a−f) and by UV−visible spectroscopy (Figure 2g, and Supporting Information). In Figure 3, we show the change of spectral absorbance of GNPs under various buffer conditions. In water, the unmodified GNPs have an absorbance peak at 530 nm.

Figure 3. (a) Spectral absorbance peak position and (b) peak width of GNPs in water, in MES, and coated with NFLt. In the presence of 160 mM NaCl, the absorbance peaks of bare GNPs shift toward higher wavelength due to aggregation whereas the peak of GNPs coated with NFLt remains unshifted in the presence of the salt, indicating that aggregation was prevented.

When the solution is exchanged to 20 mM MES, the peak absorbance is unchanged, although a slightly broader peak is observed, which could indicate the presence of a few small aggregates. However, when NaCl is added to the MES, a broad absorbance peak is measured, which is composed of a peak at 565 nm and a second overlapping peak at 652 nm. This can explain the blue color observed and the presence of aggregates due to the screening of surface charges by the salt. In contrast, when the GNPs were modified with the NFLt, the addition of 160 mM NaCl did not change the absorbance peak (Figure 4a). Further addition of NaCl, up to 2 M, did not lead to observable aggregation of the NFLt-modified GNPs (Figure 4). We do detect a slight shift of absorbance peak from 529 to 534 nm complemented with slight broadening of the peak. This suggests that NFLt-modified GNPs are further stabilized by the steric and enhanced electrostatic repulsion derived from the presence of the intrinsically disordered NFLt. Moreover, at such high NaCl concentration (i.e., 2 M) we expect the NFLt to adopt a collapsed conformation, which would allow the GNPs to come closer together, but not to irreversibly aggregate. Given the unique properties of the NFLt-modified GNPs it is important to validate that grafting occurred via the cysteine thiol groups present at their N-terminus. Alternatively, nonspecific interactions can lead the NFLt to be adsorbed onto the NPs surface, rather than binding specifically at the Nterminal and forming a brush-like structure. To differentiate between the two alternatives, NFLt-modified GNPs were placed in the presence of 150 mM NaCl and excess of DTT or 2-mercaptoethanol. Here, the GNPs quickly aggregated as a result of reducing and dissociating the surface S−Au bonds linking the NFLt to the GNPs, as in the case of unmodified GNPs (Figure 5). Quantifying the remaining NFLt left in the solution results with 69 ± 20 NFLt/GNP and average grafting density of 0.055 ± 0.016 proteins/nm2. This result is in agreement with the QCM data and the NaCN decomposition data showing highly dense tails. Importantly, in the absence of

Figure 2. Visible imaging of GNPs with and without grafted NFLt in various buffers. The color change from pink to blue for GNPs in MES and in water in the presence of 160 mM NaCl, whereas the GNPs stabilized by NFLt stay pink, indicating that modified GNPs did not aggregate. (a) GNPs in water, (b) GNPs in MES, (c) GNPs in MES stabilized by NFLt, (d) GNPs in water and 160 mM NaCl, (e) GNPs in MES and 160 mM NaCl, (f) GNPs in MES stabilized by NFLt and 160 mM NaCl, (g) UV−vis absorbance spectra of GNPs in water and in MES with and without NFLt and in absence and presence of 160 mM NaCl. The absorbance spectra of GNPs coated with NFLt proteins do not shift in the presence of NaCl. C

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Figure 4. (a) UV−visible absorbance spectral shift of GNPs coated with NFLt in the presence of various NaCl concentrations. Spectra from bottom to top are for 0, 160, 460, 870, 1340, 1550, 1740, and 2080 mM NaCl. (b) Absorbance peak position and width as a function of the NaCl concentration. Only slight increase of peak position and width are observed in the range of 0 to 2 M NaCl, showing that largescale aggregation of the GNPs is prevented by the presence of NFLt. The red line is a guide to the eye.

Figure 6. Visible imaging of NFLt-modified GNPs in the presence of (a) 0 mM, (b) 160 mM, (c) 467 mM, and (d) 910 mM KCl. The pink color remains up to 467 mM. At higher concentration, the solution turns blue and GNPs precipitate. This is shown on the absorbance spectra (e), where a broadening of the absorbance peak occurs at 467 mM KCl. At higher KCl concentrations, the broadening of the peak is more difficult to observe as the GNPs are quickly precipitating, leaving an almost clear solution.

method is thus excellent to investigate the effect of salt composition on IDPs and to study their ion-specificity. Although monovalent salt has a mild effect on inducing attraction between NFs,7,42 divalent salts do cross-link neurofilaments native supramolecular assemblies at 5 mM MgCl2.42,43 We find that for NFLt-modified GNPs, the addition of MgCl2 does induce aggregation, and the GNPs solution turns blue at concentrations starting at 10 mM (Figure 7a). The spectroscopic analysis of the NFLt-modified GNPs in the presence of MgCl2 shows that the absorption peak of the GNPs begins shifting when the MgCl2 concentration reaches 5 mM from 529 to 538 nm (Figure 7b,c). More importantly, the width of the spectral peak broadens from 82 nm in absence of MgCl2, up to 139 nm in the presence of 15 mM. This suggests the presence of a larger proportion of aggregated particles as the concentration of MgCl2 increases. Following dialysis against MES 20 mM EDTA 5 mM pH 7 modified GNPs solution with 20 mM MgCl2 reverted back to pink color as indicated by the measured spectrum (Figure 7b). This clearly demonstrates that aggregation of the modified GNPs due to Mg2+ is reversible. SAXS was used to validate the particles dimensions in solution.44 For dispersed particles in solution, the scattering intensity, I(q), directly probes the particles electron density (ρ), where I(q) ∝ |∫ ρ(r)exp(−iqr)|2. Here, the momentum transfer is q = (4π sin(θ))/λ, where 2θ is the angle between the incident and scattered beam. Several structural parameters of the studied particles can be evaluated using SAXS. For example, at low momentum transfer values, the radius of gyration (Rg), the

Figure 5. Visible light imaging showing the change in color of (a) GNP 20 in water + DTT. (b) GNP20 modified with NFLt + 150 mM NaCl, show that the modified GNPs do not aggregate in the presence of salt. (c) GNP20 modified with NFLT + 150 mM NaCl + DTT, show that DTT removes the proteins from the surface of GNPs, thus demonstrating that the S−Au bonds are the main binding mechanism for the NFLt onto GNPs.

excess NaCl, DTT did not induce aggregation in either modified, or unmodified GNPs. The added value of grafting IDPs in general, and NFLt in particular, on structured nanoparticles is the gaining of an additional degree of freedom in modulating the interparticle interactions. Surprisingly, the interaction of the NFLt-modified GNPs is ion-specific. Following the exact procedure as above, we noticed that 500 mM KCl is enough to induce considerable aggregation (Figure 6). It is indeed known that K+ ions have a chaotropic effect, compared to the kosmotropic Na+ ions.41 This is because of the larger size of the K+ ions, which causes them to be less hydrated than Na+ and allows them to come closer to the negatively charged surface of the protein, screening its overall charges and leading to aggregation. This D

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Figure 7. (a) Visible imaging of NFLt-modified GNPs in the presence of different concentrations of MgCl2. The solution turns slightly blue in the presence of 10 mM MgCl2, and precipitation occurs at 15 mM. After dialysis against 20 mM MES, 5 mM EDTA at pH 7, the solution shifts back toward red. (b) UV−vis absorbance spectra of NFLt-modified GNPs at different MgCl2 concentrations and a sample incubated with 20 mM MgCl2 then dialyzed against MES 20 mM EDTA 5 mM pH 7. (c) Change of absorbance peak position and width as a function of MgCl2 concentration. A clear shift and broadening of the absorbance peak is observed as the MgCl2 concentration increases over 5 mM.

the Guinier approximation and P(r) using indirect Fourier transform analysis (GNOM).49 We used a home-built flow cell device, composed of a quartz capillary connected on each end with thin polyether ether ketone (PEEK) tubing. This allows us to measure 20−30 μL of GNP samples and to accurately subtract the scattering obtained from the buffer from the scattering obtained from the sample in the same position. The SAXS spectra fit yields spherical particles with 8.7 ± 1.1 nm radius, without any indication for particle aggregation (Figure 8). Because of low electron density contrast of NFLt, no obvious differences between modified and unmodified GNPs can be observed (Supporting Information). Further structural support is given by environmental scanning electron microscopy (ESEM) micrographs (Figure 9), showing that very little aggregation occurs in MES for both the bare and NFLtmodified GNPs. Structural validation for the aggregation is given by SAXS in the presence of 5 mM MgCl2. The modified GNPs were mixed with MgCl2 and immediately measured for 30 min. Here, the scattering pattern yields larger Rg (16.2 nm) and increase of I(q → 0), indicating larger MW particles, which is a direct evidence for aggregation (Figure 8a,b). Moreover, indirect Fourier transform fitting yields to P(r) with a single peak in the absence of MgCl2 and to wider distributions with multiple peaks of larger dimensions in the presence of 5 mM MgCl2 (Figure 8c). Upon further incubation (above 30 min), the SAXS intensity continuously decreases due to further GNPs aggregation and precipitation within the flow-cell device. The change of behavior of the NFLt-modified GNPs at 5 mM MgCl2 is in agreement with previous results showing onsetting gelation of neurofilaments.42,43 Our results confirm that the effects previously observed were mainly due to changes of the interactions originating from the neurofilaments’ tails, and that the contribution of the neurofilaments’ core is negligible. Indeed, as well as its chaotropic41 effect compared to NaCl, the

average molecular weight (MW), and the electron density pair distribution function, P(r), can be directly evaluated.44−46 We measured the samples using a GeniX Low Divergence CuKα radiation source (Xenocs) setup with scatterless slits,47 equipped with a Pilatus 300 K detector (Dectris). Subtracted scattering profiles were analyzed with procedures implemented in the ATSAS48−50 package (Figure 8). Rg was evaluated using

Figure 8. SAXS data from NFLt modified GNPs in absence (blue squares) and in the presence of 5 mM MgCl2 (red circles). (a) Intensity plots showing the experimental data compared to a fitting for GNPs of radii 9.3 nm in the presence of 5 mM MgCl2, we can see a deviation at low q of the curve in the presence of MgCl2 indicating the presence of aggregates. Lines in (a) are fits using indirect Fourier transform with pair distribution function shown in (c). (b) Guinier plots showing the increase in Rg in the presence of MgCl2 from 8.7 to 16.2 nm. (c) Pair distribution plot showing a single size for the GNPs in absence of MgCl2 and the appearance of a second peak at larger size in the presence of MgCl2. E

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efficient screening and aggregation. Therefore, the fact that NFLt acts as a surface stabilizer is a good indication that they are indeed grafted onto the GNPs surface shifting all GNPGNP electrostatic interactions to NFLt−NFLt interactions. In summary, we demonstrated that by grafting IDPs onto GNPs, the IDPs interactions’ can be directly measured and characterized. We show that NFLt are grafted at high density and prevent aggregation in an ion-specific manner and reversible. This suggests that this method can be a powerful tool to study IDPs interactions at low protein consumption. Notably, given the NFLt grafting density as measured by QCM and the GNP concentration, our studies have been conducted with about 0.1 nanomoles of proteins that were bound through S−Au bonds to each 1 mL of GNPs (concentrations of 0.1 μM), representing ca. 95 protein molecules on each GNP. Nevertheless, the spectral shift due to aggregation in microliters of samples is easily noticeable. These new methods open the path to design tunable surface modifiers for colloidal systems and to easier studies of interactions between IDPs in solution using SAXS or optical methods.44,52−54 Nevertheless, given the polyampholytic nature of IDPs, and NF tails in particular, intratail interactions must be accounted.7,55−57 Therefore, conventional polyelectrolyte brush theories58,59 might fail to adequately describe the experimental finding and will require additional consideration.



ASSOCIATED CONTENT

S Supporting Information *

Protein expression, protein purification, QCM modification process, and UV−vis spectrometry spectra. This material is available free of charge via the Internet at http://pubs.acs.org.



AUTHOR INFORMATION

Corresponding Authors

*(S.P.) E-mail: [email protected]. *(R.B.) E-mail: [email protected]. Phone: +972-3-6408477. Fax: +972-3-6429306. Notes

The authors declare no competing financial interest.

Figure 9. ESEM images of GNPs (a) unmodified, (b) modified with NFLt, and (c) modified with NFLt showing aggregation in the presence of 10 mM MgCl2.



ACKNOWLEDGMENTS This work was supported by the Israeli Science Foundation (Grant 571/11), the European Community’s seventh Framework Programme (CIG - 293402), the Tel Aviv University Center for Nanoscience and Nanotechnology, the Abramson Center for Medical Physics, and the Sackler Institute for Biophysics at Tel Aviv University. F.P. acknowledges the support of Legacy fund (ISF). S.P. acknowledges support from the Tel Aviv University Center for Nanoscience and Nanotechnology for a postdoctoral fellowship. R.A. acknowledges support of a Marian Gertner Institute for Medical Nanosystems fellowship. We are grateful to Professor Philp Pincus, Dr. Michal Wyrsta, Dr. Yuval Reiss, and Dr. Yevgeny Berdichevsky for their help and useful discussions and guidance with the cloning and protein expression work. We also thank Micha Kornreich, Eti Malka-Gibor, Rona Shaharabani, and Guy Jacoby for their assistance in purification, measurements, and preparations.

divalent Mg2+ ions induce cross-linking of the NFLt, leading to aggregation of the GNPs. At very low salt concentration, the counterions are confined to the surface of charged GNPs. That situation, with the concentration of ions at the surface (cs) being much higher than in bulk (cb) is favorable with respect to Coulomb forces but costs counterions entropy.51 With increasing salt concentration, the difference between bulk and surface salt density minimize until Cs ∼ Cb. This reduces the entropy penalty of counterion confinement and the screened surface charges results with GNP aggregation. In the case of multivalent salts, fewer ions are needed to screen the same amount of surface charge. Therefore, entropy “cost” is smaller for divalent salt solution in comparison to monovalent one. It is therefore not surprising that multivalent salt is more efficient in screening and GNPs aggregation. By the same thought, a salt species that disrupts Hbonding of water (like chaotropic K+) costs more energy to be in bulk than confined to surface and thus screening is more efficient with K-based salts. Importantly, adding oppositely charged polyelectrolytes to the bulk would lead to even more



ABBREVIATIONS NP, nanoparticle; GNP, gold nanoparticle; IDP, intrinsically disordered protein; NF-L, H, neurofilament low, high; NFLt, F

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neurofilament low tail; PCR, polymerase chain reaction; QCM, quartz crystal microbalance; SAXS, small-angle X-ray scattering; DTT, dithiothreitol; EDTA, ethylenediaminetetraacetic acid; MES, 2-(N-morpholino)ethanesulfonic acid; SDS, sodium dodecyl sulfate; PBS, phosphate buffer saline; Rg, radius of gyration; MW, molecular weight; P(r), pair distribution function



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