Protein Entrapment in Polymeric Mesh: Diffusion in Crowded

Article pubs.acs.org/Macromolecules

Protein Entrapment in Polymeric Mesh: Diffusion in Crowded Environment with Fast Process on Short Scales Sudipta Gupta,*,†,‡ Ralf Biehl,*,§ Clemens Sill,§ Jürgen Allgaier,§ Melissa Sharp,∥,⊥ Michael Ohl,†,‡ and Dieter Richter§ †

Juelich Centre for Neutron Science (JCNS), outstation at SNS, PO Box 2008, 1 Bethel Valley Road, Oak Ridge, Tennessee 37831, United States ‡ Biology and Soft Matter Division, Neutron Sciences Directorate, Oak Ridge National Laboratory (ORNL), PO Box 2008, 1 Bethel Valley Road, Oak Ridge, Tennessee 37831, United States § Juelich Centre for Neutron Science (JCNS) and Institute for Complex Systems (ICS), Forschungszentrum Jülich GmbH, 52425 Jülich, Germany ∥ Institute Laue-Langevin (ILL), 71 rue des Martyrs, 38042 Grenoble, Cedex 9, France ⊥ European Spallation Source (ESS), PO Box 176, SE-221 00 Lund, Sweden ABSTRACT: The natural environment of proteins is a crowded environment as in cells, extracellular fluids, or during processing. Semidilute polymer solutions have been a source of rich structural and dynamical properties and mimic a crowded environment, but a proper understanding of protein dynamics in the crowded environment is far lagging. Such a study not only realizes protein’s natural environment in a crowded solution in the cell or during processing but also manifests the underlying protein−polymer interaction. By dispersing model globular proteins like α-lactalbumin (La) and hemoglobin (Hb), in aqueous solution of poly(ethylene oxide) (PEO) we mimic a crowded environment and use state-of-the-art neutron spin echo (NSE) and small-angle neutron scattering (SANS) techniques to observe the corresponding protein dynamics in semidilute polymer solution. NSE can access the fast diffusion process (Dfast) prior to the slow diffusion process on long times and length scales (Dγ). The protein dynamics in a crowded environment can be described analogous to the diffusion in a periodic potential. The fast dynamics corresponds to diffusion inside a trap built by the polymer mesh while the slower process is the long time diffusion on macroscopic length scales also observed by other techniques. We observe the onset of fractional diffusion for higher concentrated polymer solutions.

1. INTRODUCTION

techniques. PEO is also used in protein crystallization as precipitating agent.18,19 Recently, the diffusion of nanoparticles (NP) in polymeric fluids has proven to be of considerable importance in understanding the underlying dynamics of the complex fluid itself.20−22 The general question is how the diffusion on short times, before the particle feels the crowding (the polymer network), is related to the long time diffusion. In this regard globular proteins form excellent candidates to investigate the translational dynamics in a polymer network as an absolute monodisperse system. This allows us to compare with recently proposed scaling behavior of NP−polymer solution. Most of these theoretical works are based on either the hydrodynamic interactions between particles and polymers23,24 or by the “obstruction effect”25 caused by the so-called porous polymer

Many biological phenomena involve transport of proteins through media consisting of concentrated biopolymer like DNA, through polysaccharide solutions1,2 or cytoplasm, resulting in macromolecular crowding3 and chemotaxis.4 Such processes need a deeper understanding of diffusion5−7 and sedimentation8 of particles through a solution of nonadsorbing polymer chains8,9 or biomacromolecular solutions.10−13 In this connection poly(ethylene oxide) (PEO) has been proved to be an ideal model system because of biocompatibility and high solubility. PEO is used for tuning the biocompatibility of industrial materials as a surface coating polymer,14,15 as support for tissue in cartilage transplants during plastic surgery,16 and for synthesizing bioerodible photopolymerized hydrogels used to avoid inflammation or to control drug release.17 In this regard, proper knowledge about diffusion processes in macromolecular crowding is a prerequisite. Diffusion processes are also important in protein purification or in cell-fusion © XXXX American Chemical Society

Received: October 16, 2015 Revised: February 4, 2016

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DOI: 10.1021/acs.macromol.5b02281 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

using state-of-the-art neutron spin echo spectroscopy (NSE). NSE is the only method that can access the necessary time and length scales. 1.1. Fractal Diffusion in a Periodic Potential. We can view the highly entangled polymeric solution above the overlap concentration ϕ* as a mesh with polymer chains building a potential well. The potential well U(x) acts like a trap for probe particles, which has to be overcome for large-scale diffusion. In the limit of a dense mesh with an increased potential well this would result in a jump diffusional process. In this sense we use the theory of probe diffusion in a periodic soft potential. The periodicity corresponds here to the mesh size ξ(ϕ). The heterogeneity in size distribution of a polymeric mesh built by entangled polymer chains will be neglected in our model but could be included by usage of distributions describing a realistic heterogeneous network. Consequently, the physical quantities are averages. The details of the potential U(x) and the hierarchy of the polymer network on larger length scales could result in a fractional diffusion process with a fractional exponent γ ≤ 1 (γ = 1 corresponds to normal diffusion in an isotropic medium). Thus, a rough potential U(x) due to polymer chains reaching into the individual traps may induce fractional diffusion inside of the trap, while the polymer network hierarchy induces long time fractional diffusion. The fractal diffusion constant for long time Dγ is expected to depend on the shape of the potential. For overdamped Brownian motion it is given by34−36

solutions to the nanoparticles. However, once the polymers start to overlap above the overlap concentration ϕ* = 3Mw/ (4πρNARg3), these theories do not take into consideration the relaxation of the polymer segment. Here, NA is Avogadro’s number, whereas Mw, ρ, and Rg are the polymeric molecular weight, bulk density, and the radius of gyration, respectively (Table 1). Cai et al.20 recently proposed that the NP mobility is Table 1. Polymer Characterization for NSE Measurements: Weight-Averaged Molecular Weight Mw, Polydispersity Mw/ Mn, Bulk Density ρ, Radius of Gyration Rg, and the Overlap Concentration ϕ* for Two Different Samples Investigated polymer

Mw [g/mol]

Mw/Mn

ρ [g/cm3]

Rg [Å]

ϕ* [%]

dPEO d/hPEO

380000 216000

1.2 1.3

1.227 1.208

382 228

0.21 0.59

affected by the polymer dynamics. The terminal particle diffusion coefficient (Dt) scales as a power law of the particle size (d) and solution concentration (ϕ). It was assumed that the polymers are mobile, and therefore particles with size larger than the mesh size ξ(ϕ) are not frozen, but their mobility is determined by the polymer dynamics. ξ(ϕ) is defined by excluded volume or steric interaction, which prevents the polymer segments from coming arbitrarily close to each other.26 On the other hand, with increasing polymer concentration the correlation between the effective viscosity and Dt (given by Stokes−Einstein relation (SER))27 is modulated by different characteristic length scales. Recent molecular dynamics simulations28,29 have identified Rg as the important crossover length scale where Dt relates to the macroviscosity (SER applicable) rather than the local viscosity. Therefore, a corresponding crossover concentration relating the mesh size ξ(ϕ) with that of particle size (d) could be defined as ϕc = ϕ*(Rg/d)1.32.20,30,31 However, things get more exciting when globular proteins are dispersed into such polymeric solution at concentration ϕ > ϕc > ϕ* where the solution is semidilute. At such a concentration the hydrodynamic interaction between the protein and polymer starts to disappear or is partially screened at a length scale larger than ξ(ϕ) due to the presence of other chains. At the same time the protein−polymer excluded volume interaction additionally comes to play. Thereby, a competition between these two phenomena governs the system behavior in such a crowded environment. Molecular crowding can directly affect the dynamics of the protein macromolecule, manifesting an anomalous subdiffusive behavior.32 Subdiffusion is expected for particles diffusing in the presence of obstacles or binding sites building a hierarchy.32,33 In an energy landscape view this is related to energy minima where the particles are trapped building a rough energy surface. Subdiffusion is expected on length scales where a hierarchy of traps is found,33 allowing crossover between diffusion and subdiffusion dependent on the change of hierarchy. In our present work we utilize neutron scattering experimental techniques to conduct a detailed investigation of the underlying structure and dynamics of proteins as nanoparticles trapped in a polymer mesh. We propose a model to describe the diffusion of the protein NPs in a periodic potential as a result of the trap created by the polymer mesh showing subdiffusion. It allows us to describe the short time trapped diffusion and long time translational diffusion systematically with increasing polymer concentration. A detailed quantitative analysis is performed

Dγ = D0 /⟨eU / kT ⟩⟨e−U / kT ⟩

(1)

with the average over the unit cell ⟨·⟩. Here kB is Boltzmann’s constant and T is the absolute temperature. The diffusion coefficient D0 is given by the Einstein relation D0 = kBT/ζ as the self-diffusion coefficient of the particle in a solvent with friction coefficient ζ. Friction is due to interaction with the solvent but may also result from hydrodynamic interactions of the probe with the polymer chains. For polymeric solutions this could be attributed to the diffusion of a probe in a polymeric solution of same polymer density without entanglements; thus, the chain length is smaller than the entanglement length. The dynamics of a probe in a periodic potential is described by the fractional Fokker−Planck equation36,37 ∂f ∂ ⎛ ∂f ζ ∂U (x) ⎞ = D1t − γ Dγ ⎜ − ⎟ ∂t ∂x ⎝ ∂x kBT ∂x ⎠

(2)

with the periodic potential U(x) and kBT as the thermal energy. Here, the generalized fractional diffusion coefficient is given by Dγ and by the diffusion operator D1−γ as defined by West.38 An t important quantity describing diffusional processes is the meansquare displacement msd(t) = ⟨(x(t) − x(0))2⟩. For a periodic potential U(x) = (U0/kBT) cos(2π/a) with the barrier height U0 and periodicity length a, the msd is described by37 1 − Γ−1(γ + 1)Dγ ⎛ t ⎞γ msd(t ) = 2Γ−1(γ + 1)Dγ ⎜ ⎟ + 2 ⎝τ⎠ λeff × (1 − Eγ ( −λeff (t /τ )γ ))

(3)

with τ as the mean waiting time between jumps from one trap to the other to rescale the time and effective eigenvalue for normal diffusion λeff. The latter is dependent on the potential. The first component describes the long time fractional diffusion with the fractional diffusion coefficient Dγ and the Gamma B


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Macromolecules function Γ. The short time component is described by the Mittag−Leffler function (MLF) E(−atγ). Kalmykov approximates an infinite series by a single MLF function.37 The prefactor describes the msd covered by the particle while being trapped, corresponding to the fast process. For a harmonic potential with infinite potential height the prefactor describes the msd inside of the harmonic potential for γ = 1.39 Therefore, we subsume the prefactor in the following as the additional mean-square displacement inside the trap ⟨u2⟩. To further simplify, we set λeff(t/τ)γ = (λt)γ with an effective relaxation time 1/λ for the diffusion process in the trap while covering ⟨u2⟩, resulting in our msd model

with a pipet. About one-third of the solution was left in the tube to make sure that the aggregates are not stirred up and mixed with the sample again. After the centrifugation the mass of aggregates was less than 0.1% of the protein mass if they were detectable by DLS at all. This state of the samples was found to be stable at 20 °C for at least 5 days except for the appearing Hb aggregates. We choose two different batches of poly(ethylene oxide) (PEO) for NSE measurements (see Table 1): partially deuterated d/h-PEO with d/h = 4:1 to match D2O scattering contrast and fully deuterated dPEO. For SANS measurements equivalent to the matched d/h-PEO sample concentrations we used h-PEO with Mw of 443 000 g/mol. Different PEO volume fractions were prepared by dissolving the polymer in the stock protein solution at 20 °C with constant stirring for 2 days. At the same time separate PEO−D2O solutions for the same polymer volume fraction were prepared as background samples. 2.2. Neutron Scattering. We have performed small-angle neutron scattering (SANS) at the Bio-SANS instrument44 at High Flux Isotope Reactor (HFIR) source at Oak Ridge National Laboratory (ORNL) in Oak Ridge, TN. The neutron spin echo (NSE) measurements were performed at the IN-15 spectrometer at ILL, Grenoble, France. The samples in deuterated solvents were measured at 20 °C in quartz glass Hellma cells with a 2 mm path length in the neutron beam for the concentrated solutions. For the dilute solutions 4 and 5 mm path lengths were chosen to maximize the scattering intensity. In all the measurements a beam transmission above 80% was maintained to avoid multiple scattering. In NSE diffraction mode the NSE coils are used to allow polarization analysis to separate coherent and incoherent scattering comparable to polarized SANS.45 The resolution of SANS and NSE is dominated by effects from the wavelength distribution, which is around 10% for SANS instruments and 15% for IN15.46 The coherent form factor P(Q) of a single particle can be calculated from the atom positions ri and their scattering lengths bi given by

msd(t ) = 6Γ−1(γ + 1)Dγ t γ + 6⟨u 2⟩(1 − Eγ ( −(λt )γ )) (4)

For normal diffusion with γ = 1 and D = Dγ=1, the MLF simplifies to an exponential function E(−atγ) → exp(−at). For a simple diffusion process in a periodic potential we get msd(t ) = 6Dt + 6⟨u 2⟩(1 − exp( −λt ))

(5)

In the present study the factor 6 here attributes to the diffusion in 3 dimensions. Branka et al.40 used a similar function based on computer simulations of particles in a periodic potential. On Taylor’s expansion of eq 5 one can obtain a relation between the initial fast diffusion (Dfast) and the slow long time diffusion D as Dfast = D + ⟨u 2⟩λ

(6)

Because of the additional ⟨u2⟩λ, we refer in the following to Dγ as slow diffusion independently of γ and to Dfast as the fast diffusion. At short times the Smoluchowski equation41 gives a direct result for the msd as42 msd(t ) = 6D0t +

6D0 2 2T

P(Q ) =

(9)

j,k

In SANS the measured scattered intensity I(Q) is proportional to the product of the particle form factor P(Q) and the structure factor S(Q), which comprises interparticle interaction effects. For isotropic monodisperse collection of spherical particles it is given by

2 3 dF 2 6D0 ⎛⎜ dF ⎞⎟ t + t 2 + O(t 4) dx 6T 2 ⎝ dx ⎠ (7)

I(Q ) = NzΔρ2 P(Q )S(Q )

(10)

where Nz is the particle number density and Δρ is the contrast difference of the particle with respect to the solvent. In dilute concentration the structure factor is constant, S(Q) = 1. Equation 10 is proven to be useful for anisotropic but identical proteins. Hence, for proteins with a known structure (e.g., from crystallography) the form factor can be calculated with eq 9. NSE spectroscopy measures the normalized intermediate scattering function I(Q,t)/I(Q,0) as a function of Fourier time t at a given momentum transfer Q. However, because of the 2/3 probability of spin-flip for spin incoherent scattering at the NSE instrument, we measure the normalized total signal combining the coherent and the incoherent scattering as

In neutron scattering experiments the msd can be related to the intermediate scattering function I(Q,t) assuming validity of the Gaussian approximation. For a particle diffusing in three dimensions we find for the intermediate scattering function43 ⎛ 1 ⎞ I(Q , t ) = exp⎜ − Q 2⟨(x(t ) − x(0))2 ⟩⎟ ⎝ 6 ⎠

∑ ⟨bjbk exp(iQ (rj − rk))⟩

(8)

2. EXPERIMENTAL SECTION 2.1. Sample Description. The α-lactalbumin (La) and hemoglobin (Hb) proteins were bought from Sigma-Aldrich as freeze-dried (lyophilized) powder. The estimated molar mass was 14.1 kDa (α-La) and 64.5 kDa (Hb). As basis for the protein solution D2O serves as a buffer solution. Separately, a α-La and Hb stock solution was prepared with the highest needed concentration by dissolving the freeze-dried powder in the buffer solution. For α-La in D2O it corresponds to a concentration of 20 ± 0.2 mg/mL, and for that of Hb in D2O it corresponds to 18.3 ± 0.1 mg/mL. The concentration of the stock solutions were measured with the UV/vis absorption spectrometer “Thermo Nanodrop” at λ = 280 nm. The quality of both protein solutions regarding aggregates was checked with dynamic light scattering (DLS). It was found that typically 10% or more of the protein mass is aggregated. To remove these aggregates, the protein stock solution was centrifuged at a relative centrifugal force of 21000g for 2 h. The aggregates sediment to a pellet at the bottom of the centrifuge tube. The protein solution can be taken out of the tube

1

I(Q , t )/I(Q , 0) =

σcohIcoh(Q , t ) − 3 σincIinc(Q , t ) 1

σcohIcoh(Q , 0) − 3 σincIinc(Q , 0)

(11)

Here σcoh and σinc are the coherent and incoherent scattering cross section, with Icoh(Q,t) and Iinc(Q,t) corresponding to coherent and incoherent intermediate scattering function. For each Q the data are normalized by measuring graphite as the resolution sample with equal count statistics like that of the sample. Details of NSE measurement techniques can be found elsewhere.43,47,48 The background signal, here referring to any coherent and incoherent nonprotein scattering, has contributions from the solvent molecules and from the PEO network. While solvent molecules show a fast diffusion, a polymer network is expected to show at low Q collective dynamics with a crossover at Q* = 1/ξ to Zimm-like dynamics of the chains.49 For d-PEO a coherent contribution is C


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Macromolecules present, but the incoherent contribution is minimized. For D2O matched d/h-PEO the incoherent contribution is larger compared to the d-PEO because of the additional hydrogens. For matched d/hPEO the coherent network contribution vanishes for Q = 0 but may still show at larger Q a small correlation peak related to monomer position correlation of randomly deuterated/protonated monomers.50 Whatever the background contributions are, the NSE signal for the protein NPs are obtained after subtracting the corresponding background sample signal, prepared for the same concentrations of PEO in D2O solution and measured with equal statistics as that of the sample. In this way we account for any dynamics in the background resulting in the mainly coherent dynamics of the proteins in the SANS regime used here.

3. RESULTS AND DISCUSSION 3.1. Structural Characterization. Relative Size of Dynamic Entities. We implemented DLS to determine the characteristic size (d = 2Rh) of the globular proteins from their hydrodynamic radius (Rh). For α-lactalbumin (La) it yields a hydrodynamic diameter of d = 39.4 ± 1.2 Å, and for hemoglobin (Hb) protein a hydrodynamic diameter of d = 57.2 ± 1.0 Å is obtained. From DLS, the values of terminal diffusion coefficient of globular protein nanoparticles in pure solvent (Ds) are 16.5 ± 1.1 Å2 ns−1 (α-La) and 6.23 ± 1 Å2 ns−1 (Hb). The information about the hydrodynamic properties of the proteins is supplemented by calculations using the software HYDROPRO.52 The detailed procedure involves several steps described elsewhere.53 For two different set of polymer chains the radii of gyration Rg are 228 Å (d/h-PEO) and 382 Å (dPEO).54,55 The details of the polymer characterization are reported in Table 1. The ratio Mw/Mn is determined by combination of 1H NMR, size exclusion chromatography (SEC), and end-group analysis, which depicts the polydispersity of the polymer molecular weight. The theoretical crossover concentration ϕc for α-La-d/h-PEO and that for Hb-d-PEO systems in D2O solution are 2.66% and 6.05%, respectively. We have determined the concentration-dependent mesh size ξ(ϕ) for two different polymers using SANS. Figure 1a,b illustrates the measured SANS scattering intensity I(Q) for hPEO and d-PEO systems in D2O. Each of the scattering curves is normalized by the corresponding polymer volume fraction as indicated in the legends. The PEO chain mesh size ξ(ϕ) can be determined by fitting an Ornstein−Zernike (OZ) function to the SANS data, which is given by I(Q )/ϕ = I0/[Q 2ξ(ϕ)2 + 1]

Figure 1. Experimental normalized scattering intensity I/ϕ for (a) hPEO (Mw = 443 000 g/mol) at concentrations corresponding to d/hPEO and (b) for Hb-d-PEO systems in D2O solution. The solid lines are the fits using Ornstein−Zernike (OZ) approximation. (c) The concentration dependence of the mesh size ξ(ϕ) following ∼ϕ−0.76 power law for good solvent.51

α-La-d/h-PEO System in D2O. α-La neutron diffraction scattering shows the expected scattering pattern comparable to the α-La protein form factor as shown in Figure 2a. The form factor P(Q) is calculated from the atomic positions according to protein data bank (PDB) structure 1f6r. A slight coherent background is included, which is due to coherent scattering of the solvent and from the d/h-PEO. The coherent d/h-PEO contribution increases slightly with increasing concentration. The coherent d/h-PEO background for this matched polymer is a result from the composition of deuterated and protonated monomers. Matching conditions mean that in the limit Q = 0 the average contrast vanishes. Nevertheless, this condition is not satisfied at shorter length scale (larger Q) and shows a coherent correlation peak related to the average distance between monomers.50 Hb-d-PEO System in D2O. In case of the Hb-d-PEO-D2O system we observed strong impact on the formation of protein−polymer composite, which is found to be directly proportional to polymer concentration. In the following we provide experimental evidence to highlight this phenomenon, which is not found for α-La and highlights the changed interaction between protein and polymer. Figure 2 illustrates the neutron diffraction scattering intensity I(Q) for (a) ϕ = 5%, (b) 10%, and (c) 15% d-PEO in Hb-D2O solution. Here the contribution of the corresponding polymer concentration is subtracted from scattering intensity. The gradual evolution of the formation of a structure factor with increasing polymer concentration is unambiguous. To explain the data, we use a model where free proteins and nanoaggregates of proteins coexist, separated by the fraction of free protein “f ” given by

(12)

Here I0 represent the SANS forward scattering Intensity. For the h-PEO system in D2O, the measured concentrations are 2%, 10%, and 20%, which correspond the d/h-PEO system, resulting in mesh sizes of 31 ± 2, 10 ± 1, and 5 ± 1 Å, respectively. On the other hand, for the d-PEO system in D2O the measured concentrations are 5%, 10%, and 15%, which correspond to mesh sizes of 24 ± 5, 10 ± 2, and 5.2 ± 1 Å, as reported in Table 2. At low Q ( 0.09 Å−1) the excess scattering from the protein−polymer aggregates comes to play. The slow diffusion of the aggregates outside the NSE time window starts to dominate and we begin to observe a flat elastic plateau. At ϕ = 5% the system is simple diffusive with exponent γ = 1. The protein diffuses over a length scale ⟨u2⟩1/2 = 5.14 Å inside the polymer mesh with ξ = 24 Å, before it jumps onto the next mesh with an average slow diffusion Dγ. On increasing concentration from 5% to 10% to 15%, the exponent decreases to γ = 0.92, which reflects the fractional nature of the diffusion. On the other hand, the time spent by the protein in the mesh also increases from 1/λ ∼ 15 to 20 ns. Our estimation of the size-dependent crossover concentration ϕc = 2.66% also points toward the fact that the particle mobility at ϕ = 5% and above is affected by the hindrance from hydrodynamic drag between the protein and the polymer segment. The screened hydrodynamic interaction at higher concentration also decreases the effective diffusion, an effect observed in both Dfast and Dγ. It should be noted that using eq 6 we obtain Dfast = 3.69 Å2 ns−1 (5%), 2.61 Å2 ns−1 (10%), and 2.27 Å2 ns−1 (15%) (Table 2). These values are very close to that obtained experimentally (from initial slope), which proves the validity of eq 6. The small difference attributes to the fractional nature of the diffusion. The mobility of the Hb protein inside the trap also changes. But for all the concentration we observe ⟨u2⟩1/2 < ξ(ϕ), a feature different from the α-La-d/h-PEO system. The subdiffusive behavior is coupled to the hierarchy in the polymer network structure. For slow diffusion Dγ the change in hierarchy corresponds to different mesh sizes ξ with our measured mesh size as an average mesh size. On short time scales the corresponding diffusive behavior associated with ⟨u2⟩λ depends on the nature of the harmonic potential, which is also dependent on the local mesh size. As long as the potential is flat with steep walls, we should have normal diffusion in the trap, and on long time scale we should have subdiffusion because of the different mesh sizes. However, if the potential is not flat even at short time scales one can observe subdiffusive

Figure 5. Normalized effective viscosity as a function of polymer concentration for α-La-d/h-PEO-D2O and Hb-dPEO-D2O systems. The solid circles represent the effective viscosity corresponds to entanglement molecular weight of PEO chains.

It should be noted that the effective viscosity, which corresponds to Dfast, is always larger than that of the solvent viscosity ηs. For D2O it is ηs = 1.25 mPa s at 20 °C.64 On the other hand, it is always less, than the effective viscosity corresponding to the entanglement molecular weight of PEO chains (solid circles) ηeff(Me). Explicitly, this means that the simple idea of a particle inside of a mesh shows diffusion as in the solvent without polymer is wrong. The particle feels the polymer chain extending inside of a mesh, thus feeling interactions with unentangled chains through direct or hydrodynamic interaction. How strong the protein feels the polymer seems to depend on the strength of the interaction.

4. CONCLUSION The diffusion of nanoparticles in polymeric meshes is of great importance for bioprocessing but also for processing of nanoparticles for industrial applications. The crossover from short time fast dynamics to longer times until now was not examined, as most methods can only access the long times. NSE is able to access the dynamics on the necessary short times G


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(4) Tso, W.; Adler, J. Negative Chemotaxis in Escherichia Coli Negative Chemotaxis in Escherichia Coli. 1974, 118, 560−576. (5) Onyenemezu, C. N.; Gold, D.; Roman, M.; Miller, W. G. Diffusion of Polystyrene Latex in Linear Polystyrene Nonaqueous Solutions. Macromolecules 1993, 26, 3833−3837. (6) Gold, D.; Onyenemezu, C.; Miller, W. G. Effect of Solvent Quality on the Diffusion of Polystyrene Latex Spheres in Solutions of Poly(methyl Methacrylate). Macromolecules 1996, 29, 5700−5709. (7) Gupta, S.; Camargo, M.; Stellbrink, J.; Allgaier, J.; Radulescu, A.; Lindner, P.; Zaccarelli, E.; Likos, C. N.; Richter, D. Dynamic Phase Diagram of Soft Nanocolloids. Nanoscale 2015, 7, 13924. (8) Ye, X.; Tong, P.; Fetters, L. J. Transport of Probe Particles in Semidilute Polymer Solutions. Macromolecules 1998, 31, 5785−5793. (9) Omari, R. A.; Aneese, A. M.; Grabowski, C. A.; Mukhopadhyay, A. Diffusion of Nanoparticles in Semidilute and Entangled Polymer Solutions. J. Phys. Chem. B 2009, 113, 8449−8452. (10) Xu, J. Y.; Palmer, A.; Wirtz, D. Rheology and Microrheology of Semiflexible Polymer Solutions: Actin Filament Networks. Macromolecules 1998, 31, 6486−6492. (11) Chen, D. T.; Weeks, E. R.; Crocker, J. C.; Islam, M. F.; Verma, R.; Gruber, J.; Levine, a J.; Lubensky, T. C.; Yodh, a G. Rheological Microscopy: Local Mechanical Properties from Microrheology. Phys. Rev. Lett. 2003, 90, 108301. (12) Liu, J.; Gardel, M. L.; Kroy, K.; Frey, E.; Hoffman, B. D.; Crocker, J. C.; Bausch, a. R.; Weitz, D. A. Microrheology Probes Length Scale Dependent Rheology. Phys. Rev. Lett. 2006, 96, 1−4. (13) Grimaldo, M.; Roosen-Runge, F.; Zhang, F.; Seydel, T.; Schreiber, F. Diffusion and Dynamics of Gamma-Globulin in Crowded Aqueous Solutions. J. Phys. Chem. B 2014, 118, 7203−7209. (14) Hansson, K. M.; Tosatti, S.; Isaksson, J.; Wetterö, J.; Textor, M.; Lindahl, T. L.; Tengvall, P. Whole Blood Coagulation on Protein Adsorption-Resistant PEG and Peptide Functionalised PEG-Coated Titanium Surfaces. Biomaterials 2005, 26, 861−872. (15) Wu, J.; Zhao, C.; Lin, W.; Hu, R.; Wang, Q.; Chen, H.; Li, L.; Chen, S.; Zheng, J. Binding Characteristics between Polyethylene Glycol (PEG) and Proteins in Aqueous Solution. J. Mater. Chem. B 2014, 2, 2983. (16) Sims, C. D.; Butler, P. E. M.; Casanova, R.; Lee, B. T.; Randolph, M. A.; Lee, W. P. A.; Vacanti, C. A.; Yaremchuk, M. J. Injectable Cartilage Using Polyethylene Oxide Polymer Substrates. Plast. Reconstr. Surg. 1996, 98, 843−850. (17) Sawhney, A.; Pathak, C.; Hubbell, J. Bioerodible Hydrogels Based on Photopolymerized Poly (ethylene Glycol)-Co-Poly (alphaHydroxy Acid) Diacrylate Macromers. Macromolecules 1993, 26, 581− 587. (18) Polson, A.; Potgiter, G. M.; Largier, J. F.; Mears, G. E. F.; Joubert, F. J. the Fractionation of Protein Mixtures by Linear Polymers of High Molecular Weight. Biochim. Biophys. Acta, Gen. Subj. 1964, 82, 463−475. (19) Grabowski, M. J.; Brzozowski, a M.; Derewenda, Z. S.; Skarźnński, T.; Cygler, M.; Stepień, A.; Derewenda, a E. Crystallization of Human Oxyhaemoglobin from Poly(ethylene Glycol) Solutions. Biochem. J. 1978, 171, 277−279. (20) Cai, L.-H.; Panyukov, S.; Rubinstein, M. Mobility of Spherical Probe Objects in Polymer Liquids. Macromolecules 2011, 44, 7853− 7863. (21) Waigh, T. A. Microrheology of Complex Fluids. Rep. Prog. Phys. 2005, 68, 685−742. (22) Kalathi, J. T.; Yamamoto, U.; Schweizer, K. S.; Grest, G. S.; Kumar, S. K. Nanoparticle Diffusion in Polymer Nanocomposites. Phys. Rev. Lett. 2014, 112, 1−5. (23) Cukier, R. Diffusion of Brownian Spheres in Semidilute Polymer Solutions. Macromolecules 1984, 17, 252−255. (24) Ngai, K. L.; Phillies, G. D. J. Coupling Model Analysis of Polymer Dynamics in Solution: Probe Diffusion and Viscosity. J. Chem. Phys. 1996, 105, 8385−8397. (25) Amsden, B. An Obstruction-Scaling Model for Diffusion in Homogeneous Hydrogels. Macromolecules 1999, 32, 874−879.

in nanosecond range and on nanometer length scales as we have shown for two systems of proteins in a PEO network. We presented quantitative results for a crossover from fast dynamics to slow macroscopic dynamics that can be consistently described based on particle diffusion in a periodic potential. The fast diffusion process describes the diffusion inside of a mesh or trap until it jumps out of the trap and takes part in the slower diffusion process (while also being trapped again). We have successfully established the fact that the associated diffusion is not simple but fractional in nature, which is due to the heterogeneity of the polymer mesh in the bulk sample. Despite the differences in protein polymer interaction, we did not observe any coupling of the particle dynamics to the polymer dynamics. Thereby, the Q-independent 1/λ does not show Q3 or Q4 dependence as expected for Zimm or Rouse dynamics in polymer solution, and consequently the protein dynamics is not coupled to polymer chain dynamics. The fractal nature of the diffusion becomes more prominent at high Q’s and at higher polymer concentration. However, instead of the solvent viscosity, the effective viscosity due to the presence of unentangled polymer segments dominates the fast diffusion while diffusing inside of the traps. Additionally, we observed the formation of selective nanoscopic aggregates in specific hemoglobin (Hb) proteins only by tuning the polymer (PEO) concentration in the solution. The same effect is not observed for the α-lactalbumin (La)−polymer system. These aggregates may be a metastable intermediate phase for protein crystallization. To our knowledge, formation of such nanoscopic aggregates as determined by structure factor analysis S(Q) (cf. Figure 2) has not been observed before for crystallization with PEG.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (S.G.). *E-mail [email protected] (R.B.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank W. P. Hintzen, A. Wischnewski, and M. Monkenbusch for helpful discussions. We thank S. V. Pingali for helping us with the SANS experiment and G. Kali for helping with NSE measurements. We thank A. Glavic for helping in developing the necessary codes to model the data. Research conducted at Oak Ridge National Laboratory’s (ORNL) Spallation Neutron Source (SNS) and High Flux Isotope Reactor (HIFR) was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy.

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REFERENCES

(1) Roosen-Runge, F.; Hennig, M.; Zhang, F.; Jacobs, R. M. J.; Sztucki, M.; Schober, H.; Seydel, T.; Schreiber, F. Protein SelfDiffusion in Crowded Solutions. Proc. Natl. Acad. Sci. U. S. A. 2011, 108, 11815−11820. (2) Zimmerman, S. B.; Minton, A. P. Macromolecular Crowding: Biochemical, Biophysical, and Physiological Consequences 1. Annu. Rev. Biophys. Biomol. Struct. 1993, 22, 27−65. (3) Ellis, R. J.; Minton, A. P. Cell Biology: Join the Crowd. Nature 2003, 425, 27−28. H


Article

Macromolecules (26) Farnoux, B.; Boue, F.; Cotton, J. P. P.; Daoud, M.; Jannink, G.; Nierlich, M.; De Gennes, P. G. Cross-over in Polymer Solutions. J. Phys. (Paris) 1978, 39, 77−86. (27) Gupta, S.; Stellbrink, J.; Zaccarelli, E.; Likos, C. N.; Camargo, M.; Holmqvist, P.; Allgaier, J.; Willner, L.; Richter, D. Validity of the Stokes-Einstein Relation in Soft Colloids up to the Glass Transition. Phys. Rev. Lett. 2015, 115, 128302. (28) Liu, J.; Cao, D.; Zhang, L. Molecular Dynamics Study on Nanoparticle Diffusion in Polymer Melts: A Test of the StokesEinstein Law. J. Phys. Chem. C 2008, 112, 6653−6661. (29) Egorov, S. A. Anomalous Nanoparticle Diffusion in Polymer Solutions and Melts: A Mode-Coupling Theory Study. J. Chem. Phys. 2011, 134, 084903. (30) Rubinstein, M.; Colby, R. H. Polym. Phys. Oxford University Press: Oxford, 2003. (31) Milner, S. T. Predicting the Tube Diameter in Melts and Solutions. Macromolecules 2005, 38, 4929−4939. (32) Banks, D. S.; Fradin, C. Anomalous Diffusion of Proteins due to Molecular Crowding. Biophys. J. 2005, 89, 2960−2971. (33) Saxton, M. J. A Biological Interpretation of Transient Anomalous Subdiffusion. I. Qualitative Model. Biophys. J. 2007, 92, 1178−1191. (34) Brańka, A. C.; Das, A. K.; Heyes, D. M. Overdamped Brownian Motion in Periodic Symmetric Potentials. J. Chem. Phys. 2000, 113, 9911−9919. (35) Lifson, S.; Jackson, J. L. On the Self-Diffusion of Ions in a Polyelectrolyte Solution. J. Chem. Phys. 1962, 36, 2410−2414. (36) Heinsalu, E.; Patriarca, M.; Goychuk, I.; Hanggi, P. Fractional Diffusion in Periodic Potentials, 2006; p 065114. (37) Kalmykov, Y. P.; Titov, S. V.; Coffey, W. T. Fractional Diffusion in a Periodic Potential: Overdamped and Inertia Corrected Solutions for the Spectrum of the Velocity Correlation Function. Phys. Rev. E Stat. Nonlinear, Soft Matter Phys. 2012, 85, 1−9. (38) West, B.; Bologna, M.; Grigolini, P. Physics of Fractal Operators; Springer-Verlag: New York, 2003. (39) Kneller, G. R. Quasielastic Neutron Scattering and Relaxation Processes in Proteins: Analytical and Simulation-Based Models. Phys. Chem. Chem. Phys. 2005, 7, 2641−2655. (40) Brańka, a. C.; Das, a. K.; Heyes, D. M. Overdamped Brownian Motion in Periodic Symmetric Potentials. J. Chem. Phys. 2000, 113, 9911−9919. (41) Risken H.; Frank T. The Fokker−Planck Equation, 2nd ed.; Haken H., Ed.; Springer-Verlag: Berlin, 1996. (42) Tough, R. J. A.; Pusey, P. N.; Lekkerkerker, H. N. W.; Van Den Broeck, C. Stochastic Descriptions of the Dynamics of Interacting Brownian Particles. Mol. Phys. 1986, 59, 595−619. (43) Richter, D.; Monkenbusch, M.; Arbe, A.; Colmenero, J. Neutron Spin Echo in Polymer Systems. Adv. Polym. Sci. 2005, 174, 1−221. (44) Lynn, G. W.; Heller, W.; Urban, V.; Wignall, G. D.; Weiss, K.; Myles, D. A. A. Bio-SANS-A Dedicated Facility for Neutron Structural Biology at Oak Ridge National Laboratory. Phys. B 2006, 385−386, 880−882. (45) Gentile, T. R.; Jones, G. L.; Thompson, a. K.; Barker, J.; Glinka, C. J.; Hammouda, B.; Lynn, J. W. SANS Polarization Analysis with Nuclear Spin-Polarized 3 He. J. Appl. Crystallogr. 2000, 33, 771−774. (46) Pedersen, J. S.; Posselt, D.; Mortensen, K. Analytical Treatment of the Resolution Function for Small-Angle Scattering. J. Appl. Crystallogr. 1990, 23, 321−333. (47) Farago, B. Neutron Spin Echo Spectroscopy; Lecture Notes in Physics 601; Mezei, F., Pappas, C., Gutberlet, T., Eds.; SpringerVerlag: Heidelberg, 2003. (48) Monkenbusch, M. Neutron Spin Echo Spectroscopy; Lecture Notes in Physics 601; Mezei, F., Pappas, C., Gutberlet, T., Eds.; Springer-Verlag: Heidelberg, 2003. (49) Ewen, B.; Richter, D. Neutron Spin Echo Investigations on the Dynamics of Polymers. Adv. Polym. Sci. 1997, 134, 1−129. (50) Krutyeva, M.; Brás, A. R.; Antonius, W.; Hövelmann, C. H.; Poulos, A. S.; Allgaier, J.; Radulescu, A.; Lindner, P.; PyckhoutHintzen, W.; Wischnewski, A.; et al. Association Behavior, Diffusion,

and Viscosity of End-Functionalized Supramolecular Poly(ethylene Glycol) in the Melt State. Macromolecules 2015, 48, 8933−8946. (51) Gennes, P.-G. Scaling Concepts in Polymer Physics, 1st ed.; Cornell University Press: Ithaca, NY, 1979. (52) García De La Torre, J.; Amorós, D.; Ortega, A. Intrinsic Viscosity of Bead Models for Macromolecules and Nanoparticles. Eur. Biophys. J. 2010, 39, 381−388. (53) Garcia de la Torre, J.; Huertas, M. L.; Carrasco, B. Calculation of Hydrodynamic Properties of Globular Proteins from Their AtomicLevel Structure. Biophys. J. 2000, 78, 719−730. (54) Polverari, M.; van de Ven, T. G. M. Dilute Aqueous Poly(ethylene Oxide) Solutions: Clusters and Single Molecules in Thermodynamic Equilibrium. J. Phys. Chem. 1996, 100, 13687−13695. (55) Devanand, K.; Selser, J. C. Asymptotic Behavior and LongRange Interactions in Aqueous Solutions of Poly (ethylene Oxide). Macromolecules 1991, 24, 5943−5947. (56) Hammouda, B.; Ho, D.; Kline, S. SANS from Poly(ethylene oxide)/Water Systems. Macromolecules 2002, 35, 8578−8585. (57) Hammouda, B. Probing Nanoscale Structures - The SANS Toolbox, 2010. (58) Pusey, P. N.; van Megen, W. Phase Behaviour of Concentrated Suspensions of Nearly Hard Colloidal Spheres. Nature 1986, 320, 340−342. (59) McPherson, A. Crytsllization of Proteins from Polyethylene Glycol. J. Biol. Chem. 1976, 251, 6300−6303. (60) Sauter, A.; Roosen-Runge, F.; Zhang, F.; Lotze, G.; Feoktystov, A.; Jacobs, R. M. J.; Schreiber, F. On the Question of Two-Step Nucleation in Protein Crystallization. Faraday Discuss. 2015, 179, 41− 58. (61) Saxton, M. J. Anomalous Diffusion due to Obstacles: A Monte Carlo Study. Biophys. J. 1994, 66, 394−401. (62) Fetters, L. J. Physical Properties of Polymers Handbook, 2nd ed.; Mark, J. E., Ed.; Springer: Cincinnati, OH, 2007. (63) Kirincic, S. Klofutar, C. Viscosity of Aqueous Solutions of PolyEthylene Glycol. S at 298.15 K. 1999, 311−325. (64) Horita, J.; Cole, D. R. Aqueous Systems at Elevated Temperatures and Pressures; Palmer, D. A., Fernández-PriniR., Harvey, A. H., Eds.; Elsevier Ltd.: Amsterdam, 2004.

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Protein Entrapment in Polymeric Mesh: Diffusion in Crowded

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