prediction of transport properties of concentrated dis- · to date, only few works deal with the...

Journal Name

Prediction of transport properties of concentrated dis-persions of particles with competing interactions vali-dated against measurements for Lysozyme solutions

Jonas Riest,∗a,b Gerhard Nägele,†a,b Yun Liu,c,d Norman J. Wagner,d and P. DouglasGodfrind,e

In the past years, interesting static features of intermediate-range microstructural ordering inLysozyme protein solutions have been extensively explored experimentally, and explained theoret-ically based on a short-range attractive plus long-range repulsive (SALR) interaction potential. Wescrutinize for the first time the applicability of semi-analytic theoretical methods predicting diffu-sion properties and the viscosity in isotropic particle suspensions to low-salinity Lysozyme proteinsolutions. The self-consistent Zerah-Hansen (ZH) scheme is used to describe the static struc-ture factor, S(q), of Lysozyme solutions for different concentrations and temperatures obtained bysmall-angle neutron scattering. Samples belonging to the dispersed-fluid and random percolatedphases are considered. The calculated S(q)’s are the input to our calculation schemes for theshort-time hydrodynamic function H(q), and the zero-frequency viscosity η . These schemes ac-count for hydrodynamic interactions included on an approximate level. Theoretical predictions forH(q) as a function of the wavenumber q quantitatively agree with experimental results at smallprotein concentrations obtained using neutron-spin echo measurements. At higher concentra-tions, qualitative agreement is preserved although the experimental hydrodynamic functions areoverestimated. We attribute the differences for larger concentrations to translational-rotationaldiffusion coupling induced by the shape and interaction anisotropy, and the patchiness of theLysozyme particles, features not included in our globular particle model. The theoretical resultsfor η are in semi-quantitative agreement with our experimental data even at larger concentrations.We demonstrate that semi-quantitative predictions of the diffusivity and viscosity of dispersions ofglobular proteins are possible given only the equilibrium structure factor of the solution.

1 IntroductionIn the past years, the experimental and theoretical studiesof sophisticated bio-particle systems such as protein solutionshave come into the focus of soft matter science1–5. Fromthe theory viewpoint, it is tempting to take advantage of thebroad variety of well-established calculation methods describ-ing the Brownian dynamics, and in particular the equilibrium-

a Forschungszentrum Jülich GmbH, ICS-3 - Soft Condensed Matter,52428 Jülich, Germany.b Jülich-Aachen Research Alliance JARA - Soft Matter, 52425 Jülich, Germany.c Center for Neutron Research, National Institute of Standards and Technology,Gaithersburg, Maryland 20899, USA.d Department of Chemical and Biomolecular Engineering, University of Delaware,Newark, Delaware 19716, USA.e Department of Chemical Engineering, Massachusetts Institute of Technology, Cam-bridge, MA 02139, USA.∗ E-mail: [email protected]† E-mail: [email protected]

microstructure of (isotropic) hard and soft colloidal particle sys-tems (see, e.g.,6–11). These theoretical methods which have beenwell-tested and validated by comparison with experiments andcomputer simulations, are mostly based on simplifying globu-lar particle models. However, globular proteins are not ideallyspherical, having anisotropic shapes and interaction contribu-tions. Notwithstanding this fact, colloid-based calculation meth-ods were shown to apply quite well to these systems, especially re-garding equilibrium-microstructural properties such as the staticstructure factor S(q) determined in static scattering experimentsas a function of the wavenumber q. For example, there are stud-ies3,4,12–15 of the structure formation in Lysozyme protein sus-pensions where an isotropic pair potential with short-range at-tractive (SA) and long-range repulsive (LR) parts was success-fully used for calculating S(q), in good agreement with small-angle neutron scattering (SANS) data of Lysozyme solutions, inspite of the actually prolate shape of Lysozyme proteins16. Inparticular, the intermediate range microstructural order (IRO) in

Journal Name, [year], [vol.],1–10 | 1

Lysozyme suspensions, as hallmarked by a low-wavenumber peak(IRO-peak) of S(q) at a wavenumber qc has been extensively ex-plored13,17–20. Furthermore, it has been shown in recent work14

that spherical colloidal particle models are useful also for theanalysis of SANS data on flexible non-globular proteins such asmonoclonal antibodies.

To date, only few works deal with the interesting dynam-ics of SALR systems4,13,17,21. Using neutron-spin echo (NSE)measurements, it was found that the short- and long-time self-diffusion coefficients of Lysozyme solutions share roughly thesame trends13,17. Recently, the short-time dynamics of a two-Yukawa potential SALR model system has been explored theo-retically, where for the first time an IRO peak in the hydrody-namic function H(q) has been predicted22,23. Moreover, an un-expected non-monotonic temperature dependence of the meanparticle sedimentation velocity was predicted theoretically for ho-mogeneous systems. The accuracy of the semi-analytic calcula-tion methods for dynamical properties of SALR potential systemswas shown in another publication24, by comparison with state-of-the-art multi-particle collision dynamics (MPC) simulations withsolvent-mediated hydrodynamic interactions (HIs) fully included.

The isotropic SALR potential model characterizes the interac-tions and the inter-particle microstructure of Lysozyme proteinsin the one-phase dispersed-fluid phase region quite well22–24.However, it is worth to point out that the non-sphericity of theproteins and the related non-monopolar electric interaction con-tribution, as well as the surface patchiness of the Lysozyme pro-teins, can severely affect the protein dynamics at larger concen-trations. In this context, Roos et al.25 found recently in their NMRmeasurements an unusual scaling of the rotational self-diffusioncoefficient of Lysozyme solutions with the inverse of the zero-frequency viscosity η . They attribute this scaling to Lyszoyme-specific direct interactions.

In the present joint experimental-theoretical paper, we calcu-late dynamic properties of a model SALR suspension22, such asthe hydrodynamic function, H(q), the diffusion function, D(q),and the zero-frequency viscosity of Lysozyme protein solutions,and compare them with experimental results obtained by NSEand rheology measurements. The paper is organized as follows:Experimental details on the sample preparation and the SANSand NSE experiments are given in Sec. 2. In theory Sec. 3,we describe the double-Yukawa SALR potential and the methodsused for the calculation of the pair microstructure and diffusionproperties. The results Sec. 4 includes the comparison of our the-oretical results for S(q), H(q), and η with our experimental data.The summary and conclusions are given in Sec. 5.

2 Experimental Methods

2.1 Materials

Lysozyme samples were obtained from MP Biomedicals, andsubsequently purified to remove impurities and excess counter-ions to minimize the solution ionic strength. Purification wasconducted by dialyzing reconstituted Lysozyme against deion-ized water at 4◦C, until the resistance of the water reached ap-proximately 18.0 M ohm. This typically required seven changes

of deionized water over the course of 48 hours. The purifiedlysoyzme was then lyophilized by freeze drying. Samples wereprepared by dissolving purified lyophilized Lysozyme in deu-terium oxide (D2O) at 25◦C and gently vortexing to enhancedissolution and homogenization. Samples were subsequently fil-tered with 0.22µm syringe filters to remove additional impurities.Protein content was initially determined by the mass fraction ofpurified lyophilized Lysozyme in deuterium oxide, xL. The intrin-sic volume fraction of Lysozyme, φL, is then calculated accordingto the specific volume, ν0, reported in the literature (ν0 = 0.717mL/g in26), according to

φL = (xLν0)/(xLν0 +(1− xL)/ρD) , (1)

where ρD is the mass density of D2O.

2.2 SANS

SANS experiments were conducted on the D-22 and D-33 beam-lines at the Institut Laue-Langevin (ILL) in Grenoble, France, aswell as the NG-B 30 m SANS instrument at the NIST Center forNeutron Research (NCNR) in Gaithersburg, MD, following previ-ously reported protocols and methods15,17. The scattering inten-sity was obtained for scattering vector magnitudes q ranging from0.004 Å−1 to about 0.5 Å−1. All samples were held in standardquartz Hellma cells at ILL, and custom titanium cells with quartzwindows at NCNR. Low concentration samples were studied us-ing cells with a 2 mm path length to enhance intensity, while con-centrated samples were studied in cells with a 1 mm path length.All Lysozyme concentrations were studied at three temperatures(5◦C, 25◦C, and 50◦C). All raw data files were analyzed usingsoftware provided through the NCNR27. The resulting reduceddata for Lysozyme samples were fitted using an isotropic scatter-ing function.

2.3 NSE

Neutron spin echo experiments were performed on the IN-15beamline at the ILL in Grenoble, France. Samples were preparedon site following the procedure described earlier, then pipettedinto 1 mm square quartz cells and stored in a custom temper-ature controlled sample chamber. All samples were thermallyequilibrated for at least 30 minutes at each of the temperaturesstudied. For our experiments, the instrument was configured toobtain intermediate scattering functions28 at correlation times upto 50 ns at 30-35 wavenumber points ranging from q = 0.03 Å−1

to 0.20 Å−1 at each sample condition studied.

3 Theory

3.1 Interaction potential

Model interaction potentials such as the two-Yukawa13,18,19,22,29–32 and the generalized Lennard-JonesYukawa (LJY)4,12,19,24 potentials, respectively, have been widelyused for describing the diverse landscape of equilibrium andnon-equilibrium phase states in SALR systems including the dis-persed fluid4,15,19,21,22,24,33, clustered-fluid4,15,18,19,21,24,33,34,random percolated4,18,19,33, and glassy states4,17,33,35,36. In the

2 | 1–10Journal Name, [year], [vol.],

present work, we use the hard-core plus two-Yukawa (HCDY)pair potential to represent the SALR model system. The HCDYpotential reads explicitly

βV (x) =

{∞ , x < 1

−K1e−z1(x−1)

x +K2e−z2(x−1)

x , x≥ 1 ,(2)

where x = r/σ is the inter-particle center-to-center distance, r, inunits of the particle diameter σ and β = 1/kBT . Moreover, z1 andz2 determine the range of the attractive and repulsive Yukawa po-tential parts in units of σ , respectively, and K1 and K2 are the re-spective short-range attractive and long-range repulsive potentialstrengths in units of kBT . The depth of the attractive well is givenby βV (x = 1+) = K2 −K1. Theoretical results for S(q) obtainedfrom self-consistent ZH integral equation theory and Monte-Carlo(MC) simulations using the effective pair potential in Eq. (2) havebeen shown to be in good agreement with SANS data for S(q) inlow-salinity Lysozyme solutions, for systems in the dispersed-fluidphase where monomers are the dominant species.4,12,13,33.

3.2 Equilibrium microstructure

For the calculation of equilibrium pair functions of SALR systemsusing liquid-state integral equation theory methods, the impor-tance of using a self-consistent closure relation was shown re-cently in37. A self-consistent hybrid scheme for systems withattractive and repulsive pair potential parts is used here basedon the method proposed by Zerah and Hansen (ZH)8. TheZH scheme interpolates between the hypernetted chain closure(HNC) for long, and the soft-core mean spherical approximation(SMSA) for short particle pair distances r.

The ZH scheme is particularly suited for systems with soft-corerepulsion and an attractive interaction part8,38,39. To describesuch SALR systems theoretically, we split the total pair potentialV (r) into a reference part, V1(r), and a perturbation part V2(r),selected here as

V1(r) =

∞ , r < σ

V (rshift) , r ≤ rshift

V (r) , r > rshift .

(3)

and

V2(r) =

{V (r)−V (rshift) , σ ≤ r ≤ rshift

0 , r > rshift ,(4)

respectively, where rshift = rmax > σ , and where rmax is the pairdistance at which V (r) is at its maximum. Accordingly, V2(r) ispurely attractive, while V1(r) is purely repulsive (c.f. Ref.23). TheZH closure reads

g(r)≈ e−βV1(r)

[1+

e f (r)[h(r)−c(r)−βV2(r)]−1f (r)

], (5)

with the mixing function

f (r) = 1− e−ϑr . (6)

Here, the mixing parameter ϑ is determined self-consistentlyfrom enforcing equality of the isothermal compressibilities de-rived using the virial pressure and compressibility routes, respec-tively. In taking the concentration derivative of the virial pres-sure, we assume for simplicity that the mixing parameter is den-sity independent. This approximation is justified, since ϑ is only aweakly varying function of the volume fraction φ 8. In the limit ofϑ → ∞, the ZH closure reduces to the hypernetted chain closure(HNC), while in the opposite limit ϑ → 0, the soft MSA closurerelation given by

g(r)≈ e−βV1(r) [1+h(r)− c(r)−βV2(r)] (7)

is recovered.

Since there exist different choices for V1(r) and V2(r), the com-parison with simulation data for the radial distribution function,g(r), is a necessary prerequisite to assess the accuracy of the ZHscheme8. Our splitting of V (r) gives for the dispersed-fluid phaseZH based results for g(r) in excellent agreement with computersimulation predictions based on the MC, MD and MPC simulationmethods, as it is shown in Refs.22–24.

3.3 Short-time diffusion properties

Short-time diffusion in colloidal suspensions is commonly as-sessed experimentally by measuring the intermediate scatteringfunction S(q, t), where

S(q, t� τd) = S(q) exp[−q2D(q)t

], (8)

is an exponentially decaying function for correlation times, t,small compared to the structural relaxation time τd = a2/d0. Here,d0 is the Stokes-Einstein single-particle translational diffusion co-efficient for a spherical particle (protein) of radius a = σ/2 ina solvent of viscosity η0, given by d0 = (kBT )/(6πη0a) for sticksurface boundary conditions. For t� τd, the configuration of par-ticles is hardly changed by diffusion, and the particle dynamics isinfluenced solely by the quasi-instantaneously transmitted hydro-dynamic interactions (HIs). In Eq. (8), D(q) = d0 H(q)/S(q) de-notes the short-time diffusion function, and H(q) is the positive-valued hydrodynamic function. The latter characterizes the influ-ence of HIs on short-time diffusion, and it can be calculated inoverdamped Brownian dynamics starting from

H(q) = lim∞

⟨1

Nµ0

N

∑i, j=1

q̂qq ·µµµ i j(X) · q̂qqeiq·(ri−r j)

⟩eq

. (9)

Here, 〈. . .〉eq denotes an equilibrium ensemble average, and thethermodynamic limit, denoted by lim∞, is taken to describe amacroscopically large scattering volume. The µµµ i j(X) are the pro-tein configuration (i.e. X = {rrr1, . . . ,rrrN}) dependent mobility ma-trix tensor elements linearly relating the hydrodynamic force ona particle j to the velocity change of a particle i owing to thesolvent-mediated HIs. Moreover, kBT µ0 = d0 and q̂qq = qqq/q. The


diagonal terms in Eq. (9) for which i = j give the wavenumber-independent short-time self-diffusion coefficient ds, while the off-diagonal (i 6= j) terms sum up to the wavenumber-dependent dis-tinct hydrodynamic function part, Hd(q), of H(q), that character-izes the hydrodynamic force-velocity cross-correlations.

For the calculation of H(q), we use the so-called hybrid BM-PAscheme which combines the well-established Beenakker-Mazur(BM) effective medium method10,11, where many-body HIs areapproximately included for the calculation of Hd(q), with thePairwise Additivity (PA) approximation of HIs used for calculationof the self-part, ds, of H(q). In recent work by some of the presentauthors24, the accuracy of the BM-PA scheme for the calcula-tion of short-time diffusion properties of SALR systems has beenshown by the comparison with elaborate multi-particle collisiondynamics (MPC) simulation results where HIs are fully accountedfor. For details about the employed BM-PA hybrid method, we re-fer to Refs.22–24,40. Importantly, for a given set of HCDY potentialparameters, solvent viscosity, and protein concentration, the BM-PA theory is predictive and requires no fitting to the data. As it isnot possible to predict the HCDY potential parameters a priori forLysozyme in solution, we determine these parameters by fittingour SANS data. With the fitted potential parameters, we calculateand predict the dynamic properties for concentrated Lysozyme so-lutions and compare these with the experimental data.

4 ResultsIn this section, we compare the experimental results of static anddynamic properties of Lysozyme solutions with our theoreticalcalculations based on the isotropic two-Yukawa SALR pair poten-tial in Eq. (2).

4.1 Static properties

In order to explore the influence of SALR interactions on short-time and long-time dynamic properties of Lysozyme solutions,the strength and range of the competing interactions betweenLysozyme proteins are quantified using small angle scattering(SANS) measurements. Seven different solution conditions,shown in Table 1, are investigated as a function of the temper-ature and protein concentration. Under each condition, the ab-solute scattering intensities are normalized by a 1% mass fractionsample representative of the protein form factor as well as otherpre-factors to extract the inter-particle structure factor, S(q). Theresulting effective structure factors (symbols) are replotted fromprior work41 in Figs. 2a and 2c. The absolute SANS scatteringintensities were fitted theoretically for every solution conditionusing q-values between 0.02 Å−1 and 0.50 Å−1. This avoids small-q artifacts due to the beam stop and residual impurities remainingin solution after the purification and filtering, and it allows for ac-curately capturing the incoherent scattering background at largeq-values. In our theoretical model, the proteins are describedas spheres with hard-core diameter σ ≈ 30.74Å. The interactionparameters for the HCDY pair potential, {K1,K2,z1,z2}, and thevolume fraction φ presented in Table 1 were deduced from theexperimental structure factor fits employing the ZH integral equa-tion scheme with the static decoupling approximation42 used to

capture form factor and inter-protein correlation contributions tothe scattering data. In this way, we implicitly account for effectssuch as the variation of the solution pH with changing φ .

For each sample, we have investigated its individual phase stateby analyzing MC simulation data based on the HCDY interactionpotential in Eq. (2), with the potential parameters listed in Table1, and in accord with a recently proposed generalized phase dia-gram of SALR systems19. Based on these simulation results, thephase states of the samples 1-7 were determined by calculatingthe cluster size distribution function (CSD),

N(s) =⟨

sn(s)Np

⟩, (10)

defined in Ref18. The CSD characterizes the average fraction ofparticles, N(s), that are members of a cluster of size s. Accord-ingly, 〈. . .〉 denotes here an average over representative configu-rations of the simulation runs, and n(s) is the number of clustersof size s within a specific configuration where clusters are definedstructurally according to their connectivity via an effective con-tact or cut-off distance19. Moreover, Np is the total number of

particles in the system such that ∑Nps=1 N(s) = 1.

In Fig. 1, we show the CSD’s of samples 1-7. While samples1-4, and sample 7 have a monotonically decreasing N(s) charac-teristic of the monomer-dominated dispersed-fluid phase whereonly few and highly-transient clusters are present in the system,samples 5 and 6 are in a random percolated phase state. Theirrespective CSD’s have a characteristic peak at the total particlenumber (Np = 1728 particles) in the basic simulation box.

1 10 100 1000s

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

N(s

)

Sample 1Sample 2Sample 3Sample 4Sample 5Sample 6Sample 7

Fig. 1 Cluster size distribution function, N(s), of samples 1-7, obtainedfrom NVT-MC simulations for Np = 1728 particles interacting by thehard-core double-Yukawa potential in Eq. (2). According to the statediagram for this HCDY system given in 41 (see also 35), samples 5 and 6marked by filled circles are in a random percolated state 17.

While the effect of changing T and φ on the interaction parame-ters has been demonstrated in more detail elsewhere17,41, it is in-teresting to discuss here some pertinent trends related to the dataprovided in Table 1. All fitted volume fractions φ agree well withthe predictions using the intrinsic volume fraction of LysozymeφL, which are 0.0399 (5 %wt), 0.1646 (20 %wt), and 0.207 (25%wt). The range of repulsion decreases (z2 increases) with in-creasing protein concentration as expected due to the greaterionic strength caused by more counter-ions dissociated into so-


Table 1 Temperature, T , concentration (in wt [%]) and intrinsic volume fraction φL (c.f. Eq. (1)) of the probed Lysozyme in D2O samples. Additionallyshown are the employed parameter values of the HCDY potential V (r) in Eq. (2), the volume fraction φ determined from the ZH-fit of the experimentalS(q) and used in our dynamic calculations, and the fraction (in %) of the MC generated representative configurations having a percolated cluster(rightmost column). Note that samples 5 and 6 marked in grey are in a random percolated state 41

Sample T [◦C] wt [%] φL φ K1 K2 z1 z2 %1 25 5 0.0399 0.0398 6.0291 4.2743 10 1.2473 02 5 20 0.1646 0.1432 6.4666 3.2868 10 2.7839 0.63 25 20 0.1646 0.1472 5.8511 3.5588 10 2.9338 04 50 20 0.1646 0.1551 6.1753 4.3252 10 3.3055 05 5 25 0.2070 0.2017 6.3 3.0811 10 3.6117 1006 25 25 0.2070 0.2099 5.743 3.3574 10 3.8785 977 50 25 0.2070 0.2091 5.2251 3.7215 10 4.0331 0

0 2 4 6 8 10qσ

0

0.5

1

1.5

S(q)

Sample 1Sample 3Sample 7

(a)

1 1.5 2 2.5r/σ

0

0.5

1

1.5

g(r)

φ = 0.0398φ = 0.1472φ = 0.2091

1 1.5 2 2.5r/σ

0

0.5

1

1.5

g(r)

ZHMC

(b)

0 2 4 6 8 10qσ

0

0.5

1

1.5

S(q)

Sample 5Sample 6

(c)

1 1.5 2 2.50

1

2

3Sample 5Sample 6

1 1.5 2 2.5r/σ

0

1

2

3

g(r)

ZHMC

(d)

Fig. 2 (a) Comparison of the experimental (SANS) S(q) of the dispersed-fluid phase Lysozyme solution samples 1 (black symbols), 3 (red symbols),and 7 (blue symbols) in D2O with the ZH-S(q) (lines) obtained using the potential parameters in Table 1. For better visibility, the results for thedispersed-fluid phase systems 2 and 4 are not shown here. (b): ZH-generated g(r) of samples 1, 3, and 7 in comparison with corresponding NVT-MCsimulation data using Np = 4096 particles. (c) and (d): Same as (a) and (b), but now the randomly percolated samples 5 and 6 are considered. Colorand symbol codes as in Fig. 1.


lution. The attraction well depth, (K1−K2), increases with de-creasing temperature, in agreement with the expected behaviorfor protein interactions43. Interestingly, the well depth is approx-imately constant for varying φ at each temperature. The man-ifestation of these competing interaction features on the staticproperties of the Lysozyme solutions are visualized in Figs. 2aand 2c, where we show our SANS results of S(q) for Lysozymeproteins in deuterium oxide (D2O) at the selected concentrationsand temperature values T listed in Table 1. The q-range probedin the SANS measurements is constrained to values below thenext neighbor peak position, qm, of S(q). We focus first on Fig.2a where only the samples in the dispersed-fluid phase state areconsidered. The experimental S(q)’s of samples 1 and 3 have anIRO peak at qcσ . 3, while for sample 7 no clear low-q peak is vis-ible. This observation is representative of the loss of intermediaterange order due to the weaker attraction strength at higher tem-perature where the electrostatic repulsion dominates the inter-protein interaction. Interestingly, samples 3 and 7 have similarMC generated cluster size distributions, as it can be observed inFig. 1. Hence, the differences in the structure factors arise fromthe different local arrangements of proteins at the intermediaterange length scale. Note here, however, that the dominant speciesin the dispersed-fluid state are monomers plus loosely connectedsmall clusters. The agreement between the ZH-generated curvesof S(q), with potential parameters given in Table 1, and the exper-imental data is good for the systems in the dispersed-fluid phasestate, with larger deviations only visible at larger q values in theregion around qm. However, the statistical error bars in the SANSdata are quite large. Note additionally the high accuracy of theZH-calculated radial distribution functions, g(r), demonstrated inFig. 2b by the comparison with our MC simulation data for g(r).

Fig. 2c shows the experimental S(q) of samples 5 and 6 to-gether with the fitting results. These samples belong to the ran-dom percolated state according to their CSD’s in Fig. 1, and thestate-diagram discussed in Ref.41. For not too large concentra-tions in the random percolated phase state, the system is perco-lated but individual particles are still mobile both on a local andmacroscopic scale such that Newtonian flow is observed in thelong-time regime t � τd. The experimental S(q) of sample 5 hasa pronounced IRO peak, whereas for sample 6 only the onset ofa low-q peak is visible. Interestingly enough, even though sam-ples 5 and 6 are percolated their pair structure functions g(r) andS(q) are well described by the ZH integral equation results. This isshown in Fig. 2 (c) where the ZH-S(q)’s are compared to the SANSdata. According to Fig. 2 (d), the ZH-generated g(r)’s are in verygood agreement with the MC results, with only slight deviationsobservable in the reduced distance range 1.7 . r/σ . 2. This isin spite of the fact that the MC-generated CSD’s of samples 5 and6 are distinctly different from those of the monomer-dominateddispersed-fluid phase region which are monotonically decaying.

Before continuing with the discussion of dynamic properties,it is worth to correlate once again the features of the S(q) func-tions shown in Figs. 2a and 2c with the corresponding N(s) datain Fig. 1. Here, we point out that the shown N(s) curves quan-tify a time-averaged (static) cluster size distribution, which maydiffer from the size distribution of long-lived clusters containing

proteins that are dynamically correlated. However, quantifyingthe time-averaged local structure of particles in SALR systemsprovides context information for the resulting dynamical prop-erties. In particular, the local cluster size and structure, and thecluster orientation will influence the short time dynamics by re-stricting local structural rearrangement. As an extreme example,a recent study17 reported the onset of locally glassy dynamicsdue to the formation of regions of high density within a locallyheterogeneous solution structure induced by intermediate rangeorder, found to occur at elevated volume fractions of Lysozyme.In contrast, sample 1, e.g., is composed primarily of monomers inequilibrium with small clusters. As noted above, sample 3 and 7have similar cluster size distributions despite a significant differ-ence in protein interactions and concentration. However, due tothe larger concentration of sample 7, structural rearrangementsare expected to be slower and more restricted than those in sam-ple 3. This will be discussed in the next section.

4.2 Short-time diffusion properties

0 2 4 6 8 10qσ

0

0.2

0.4

0.6

0.8

1H

(q)


Fig. 3 NSE results (circles) for the hydrodynamic function H(q) of theLysozyme samples in D2O listed in Table 1. The H(q)’s are deducedfrom the experimental D(q)’s and S(q)’s using H(q) = S(q) ·D(q)/d0. Forcomparison, our BM-PA results for H(q) are included based on the ZHinput for g(r) and S(q) (solid lines). The parameters of the HCDYpotential are given in Table 1. Filled circles refer to the randomlypercolated samples 5 and 6. Same color and symbol codes as in Fig. 1.

In addition to a largely different accessible wavenumber range,an advantage of the NSE method in comparison to dynamic lightscattering (DLS) is that the scattering signal, and thus S(q, t), isnot dominated by contributions from small amounts of large sizeimpurities or aggregates, as it is in DLS. The comparison of theNSE-determined H(q) of the Lyszoyme solutions with our BM-PAtheory predictions based on the ZH-calculated g(r) and S(q) ispresented in Fig. 3. The experimental q-range is constrained hereto the IRO peak region.

Our theoretical results for H(q) in Fig. 3 are overall in de-cent agreement with the experimental data. For sample 1 havingthe lowest concentration of φ = 0.0398, the agreement is nearlyquantitative. Notice in particular the distinct IRO peak in the ex-


perimental H(q) that nicely confirms our earlier theoretical pre-dictions of this peak. With increasing φ , the predictions are qual-itatively correct but deviate quantitatively. The BM-PA calculatedH(q)’s reflect the trends of the experimental data, such as thecrossing points of the different curves, and this even for the ran-domly percolated samples 5 and 6 and the high-concentrationsample 7. We emphasize that the theoretical results for H(q) in-voke no adjustable parameter, being fully determined by the S(q)and corresponding g(r) inputs in Fig. 2. The vertical offset be-tween the experimental and theoretical H(q)’s at larger φ can bepartially attributed to the inaccuracy of the PA scheme for cal-culating the ds-part of H(q), as for larger φ the theory tends tooverestimate the slowing influence of the HIs on self-diffusion.Yet, in a recent paper by part of the authors24, the good accuracyof the hybrid BM-PA scheme for SALR systems is shown to per-sist in comparison with elaborate MPC simulations for φ valuesextending at least up to φ ≈ 0.1.

More influential to the differences between the theoretical pre-dictions and the experimental results for H(q), visible in Fig. 3,are very likely the theoretical simplifications neglecting the ac-tually non-spherical shape and interaction contributions, and thepatchiness of the Lysozyme proteins. In contrast, the S(q)’s arewell described by the spherical HCDY model even for the largerφ values, and even for the two systems considered in the randompercolated state, as it is visible in Fig. 2a and noticed in4,12,13,33.This is due to the equilibrium orientational averaging invoked inS(q). In contrast, the protein asphericities have a significantlystronger dynamic influence for larger φ , owing to the coupling oftranslational and rotational protein motion even at short times.This dynamic coupling slows the diffusion, which explains theoverestimation of the experimental H(q) by the theoretical predic-tions based on the isotropic pair potential in Eq. (2). In this con-text, using computer simulations Bucciarelli et al.44 recently re-vealed a strong slowing effect of directional attractive interactionson the cage-diffusion coefficient, D(qm), as compared to purelyisotropic attractive interactions of comparable strength. More-over, Roos et al. in25 find experimentally for Lysozyme solutionsthat the reduced rotational self-diffusion coefficient, d0

r /dr, withd0

r denoting the rotational diffusion coefficient, dr, at infinite di-lution, is approximately equal to the inverse of the reduced zero-frequency viscosity, η/η0, and to the reduced long-time transla-tional self-diffusion coefficient dL/d0. This approximate equalityis suggestive of strong correlations in the dynamics of neighbor-ing particles due to rotational self-diffusion25. Such a peculiarbehavior is not observed in globular suspensions of charged col-loidal particles where 1/dr has a weaker φ -dependence than η

and 1/dL45.

Lyszoyme-specific interactions may play an important role forthe solution dynamics. Roos et al.25 do not find the dr ∝ 1/η

scaling for proteins different from Lysozyme, notably here αβ -crystalline and bovine serum albumin proteins. An additional in-crement to the overestimation of the experimental H(q) by ourtheoretical model is possibly the presence of protein hydrationlayers resulting in enlarged hydrodynamic protein radii. How-ever, we expect the hydration layer effect to be too small to sig-nificantly account for the differences observed in Fig. 3. In sum-

mary, the shape and interaction anisotropies very likely severelyinfluence the Lysozyme dynamics at larger concentrations on thenanosecond scale, whereas the microstructure as quantified byS(q) and g(r) is less strongly affected.

0 2 4 6 8 10qσ

0

0.5

1

1.5

2

D(q

) / d

0


Fig. 4 NSE results for the experimentally directly accessible diffusionfunction, D(q) (circles), in comparison with theoretical predictions (solidlines). The theoretical D(q) is obtained from dividing the BM-PAcalculated H(q) in Fig. 3 by the respective ZH-S(q) in Fig. 2a. Samecolor and symbols codes as in Fig. 1.

In Figure 4, we compare the NSE results for the experimen-tally directly obtained short-time diffusion function, D(q), withour theoretical predictions. The agreement between theory andexperiment is of similar quality as that for H(q) in Fig. 3. In viewof Figs. 2a and 4, and owing to the small values of S(q) for lowq values, the deviations in the H(q)’s between theory and exper-iment are somewhat enhanced after the division by S(q). Quiteimportantly, from the comparison in Figure 4 it follows that in or-der to deduce ds using NSE, a broader q-range should be probed,stretching out to q-values larger than qm. However, the oscilla-tions both in the experimental and theoretical D(q)’s are small forq & qc, with the statistical errors in the NSE-D(q) for larger q be-ing comparable in magnitude to the amplitude of the theoreticalD(q) at these wavenumbers. In fact, earlier NSE measurementson Lysozyme solutions covering larger q-values revealed basicallya flat plateau of D(q) at large wavenumbers15,21. Hence, weexpect the inaccuracy in inferring ds from the large-q extrapo-lation of low-q experimental data using ds/d0 = D(q→ ∞), to bequite small. We finally note that the distribution of the momen-tary orientations of non-spherical particles (i.e., the orientationalpolydispersity) can affect the measured (effective) S(q) and D(q)similarly as size polydispersity, causing damping of large-q oscil-lations46,47.

4.3 Long-time dynamicsIn addition to short-time diffusion properties, we have deter-mined theoretically the zero-frequency viscosity η =η∞+∆η . Thepredictions for η are compared here with earlier measurementsof the Lysozyme solution viscosity taken from17,33. In contrastto the high-frequency viscosity part, η∞, of purely hydrodynamic


origin for which quite accurate analytic tools for its calculationare available, the calculation of the shear relaxation part, ∆η ,is more demanding because one needs to account for the shear-induced deformation of next-neighbor cages influenced by directand hydrodynamic interactions alike. To calculate ∆η , we usea simplified mode-coupling theory (MCT) expression where theviscosity is obtained in a first iteration step of the self-consistentMCT equations by coupling η to the dynamics of S(q, t). Explicitly,we use the expression40,48

∆η

η0

∣∣∣∣(1)MCT≈ 1

40π

∫∞

0dy y2 (S

′(y))2

S(y)1

H(y), (11)

where y = qσ and S′(y) = dS(y)/dy. In this expression, the con-tribution of the HIs to the MCT shear relaxation vertex functionis omitted, on arguing that the associated hydrodynamic mobil-ity tensors relating shear strain to hydrodynamic particle forcedipoles (stresslets) are rather short-ranged (see, e.g.,49). In Eq.(11), HIs enters only through S(q, t), approximated by its short-time form

S(y, t)≈ S(y)exp[−y2 H(y)

S(y)t

4τd

], (12)

valid for t� τd. Consequently, the relaxation part ∆η is underes-timated by the first-order MCT as compared to that by the fullyself-consistent MCT. This underestimation is expected to becomemore pronounced at larger φ .

For the high-frequency viscosity contribution, η∞, to η we usethe PA method where two-body HIs contributions including lubri-cation forces are included. See22,23,50 for details on this method.

0 0.05 0.1 0.15 0.2 0.25φL

1

2

3

4

5

η/η 0

Exp.Theory

0 0.05 0.1 0.15 0.2 0.25φL

1

2

3

4

5

5 °C25 °C50 °C

0 0.1 0.2 1

1.5

2

η ∞ /

η 0

Fig. 5 Experimental data for the normalized zero-frequency viscosity,η/η0 (circles), as function of the Lysozyme intrinsic volume fraction φL(c.f. Eq. (1) and Table 1), and compared with first-iteration MCT-PAresults (squares). In addition, the PA-calculated normalizedhigh-frequency viscosity, η∞/η0 (triangles), is shown in the inset. Notethat ∆η = η−η∞ is the shear relaxation viscosity part. In the presentfigure, a few systems additionally to the ones discussed regarding S(q)and H(q) are included. Experimental data for η are taken from 17,33. Thesamples in the random percolated phase state for which φL & 0.2 andT . 50◦C are labeled by filled symbols, while for the samples in thedispersed-fluid phase open symbols are used.

Interestingly, as shown in Fig. 5, our hybrid MCT-PA method

results for η are in decent agreement with the experimental vis-cosity measurements, in particular at smaller φ . From our PAmethod theoretical results for η∞ depicted in the inset of Fig. 5,and in view of ∆η = η−η∞, we see that the increase of η with in-creasing φ is mainly due to the shear relaxation viscosity part ∆η .We further note that η∞ is quite insensitive to changes in the pairpotential caused by temperature variations. This is in line withthe general observation that ∆η is more sensitive to the interac-tion parameters than η∞

50. As one expects, the viscosity is largestfor the lowest considered temperature. The first-order MCT ex-pression in Eq. (11) combined with the PA expression for η∞ isthus a valuable and easy-to-implement tool for the calculation ofη , giving results in reasonably good agreement with experimentalviscosity data.

5 Summary and ConclusionsWe have assessed the applicability to Lysozyme protein solutionsof our semi-analytic methods of calculating static and dynamicproperties of globular particle dispersions. These methods di-rectly relate the bulk diffusion and viscosity to the structure factorwithout additional inputs. We have compared our theoretical pre-dictions with SANS scattering data for S(q), NSE measurementsof D(q), and zero-frequency experimental viscosity data for η .Consistent with previous studies on Lysozyme solutions4,12,13,33,we find that the experimental S(q) and the inferred g(r) are over-all well described by a spherical particle model with an isotropicHCDY potential.

The quantitative agreement between the BM-PA H(q) and theexperimental data (SANS/NSE) observed at lower concentrationsis lost at larger φ , but qualitative agreement is still maintained.The experimental-theoretical deviations in D(q) and H(q) canbe attributed to the theoretically neglected asphericities in par-ticle shape and associated electrostatic interactions, as well as tothe disregarded patchiness of the short-range attraction contribu-tion, which for larger concentrations lead to a strong coupling ofthe rotational and translational particle dynamics. Our findingsare in line with earlier experimental work25 on Lysozyme solu-tions where a dr ∝ 1/η scaling is observed for Lysozyme solutionsbut not for spherical colloidal particles with isotropic interac-tions. The strong influence of particle surface patchiness on short-time diffusion was highlighted recently in a joint experimental-simulation study of the cage-diffusion coefficient D(qm) of γB-crystallin44. The authors of this study show that patchy attractiveinteractions give rise to smaller values of D(qm) than those ob-served for isotropic attractive interactions of comparable strength.

Quite unexpectedly, our theoretical predictions for the zero-frequency viscosity η are in reasonably good agreement with rhe-ological data for Lysozyme solutions, showing that the employedsimplified MCT-PA method is an efficient and easy-to-implementtool for viscosity calculations of globular protein solutions evenwith pronounced intermediate range order. Lysozyme-specificanisotropic interactions, disregarded in our spherical model, areseen to be more influential on short-time diffusion properties thanon the shear viscosity, which includes also long-time shear relax-ation contributions.

A clear distinction and quantification of non-sphericity and


patchiness effects on the protein dynamics has to await future dy-namic simulations and theoretical work where these effects willbe included. Our plan is to perform such simulations and to ob-tain benchmark results for future theoretical work.

AcknowledgmentsJ.R. acknowledges support by the International Helmholtz Re-search School on Biophysics and Soft Matter (IHRS BioSoft).G.N. and J.R. thank R.G. Winkler, S. Das (all FZ Jülich), R.Roa (Helmholtz-Zentrum Berlin) and M. Heinen (Universidad deGuanajuato, Leon, Mexico) for many helpful discussions. Thismanuscript was prepared under the partial support of the coop-erative agreements 70NANB12H239 and 70NANB10H256 fromNIST, U.S. Department of Commerce. This work utilized facili-ties supported in part by the National Science Foundation underAgreement No. DMR-1508249. Certain commercial equipment,instruments, or materials are identified in this document. Suchidentification does not imply recommendation or endorsement bythe National Institute of Standards and Technology nor does it im-ply that the products identified are necessarily the best availablefor the purpose.

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