Dynamics in Crowded Environments: Is Non-Gaussian Brownian

Apr 29, 2014 - Dynamics in Crowded Environments: Is Non-Gaussian Brownian. Diffusion Normal? Gyemin Kwon,. †. Bong June Sung,*. ,† and Arun Yethir...
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Dynamics in Crowded Environments: Is Non-Gaussian Brownian Diffusion Normal? Gyemin Kwon,† Bong June Sung,*,† and Arun Yethiraj*,‡ †

Department of Chemistry and Institute for Biological Interfaces, Sogang University, Seoul 121-742, Republic of Korea Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, United States

‡

ABSTRACT: The dynamics of colloids and proteins in dense suspensions is of fundamental importance, from a standpoint of understanding the biophysics of proteins in the cytoplasm and for the many interesting physical phenomena in colloidal dispersions. Recent experiments and simulations have raised questions about our understanding of the dynamics of these systems. Experiments on vesicles in nematic fluids and colloids in an actin network have shown that the dynamics of particles can be “non-Gaussian”; that is, the self-part of the van Hove correlation function, Gs(r,t), is an exponential rather than Gaussian function of r, in regimes where the mean-square displacement is linear in t. It is usually assumed that a linear meansquare displacement implies a Gaussian Gs(r,t). In a different result, simulations of a mixture of proteins, aimed at mimicking the cytoplasm of Escherichia coli, have shown that hydrodynamic interactions (HI) play a key role in slowing down the dynamics of proteins in concentrated (relative to dilute) solutions. In this work, we study a simple system, a dilute tracer colloidal particle immersed in a concentrated solution of larger spheres, using simulations with and without HI. The simulations reproduce the non-Gaussian Brownian diffusion of the tracer, implying that this behavior is a general feature of colloidal dynamics and is a consequence of local heterogeneities on intermediate time scales. Although HI results in a lower diffusion constant, Gs(r,t) is very similar to and without HI, provided they are compared at the same value of the mean-square displacement.

I. INTRODUCTION The dynamics of macromolecules in complex environments is of fundamental, biological, and technological importance. From a fundamental standpoint, there has been recent discussion regarding Brownian particles with linear mean-square displacement but a non-Gaussian distribution of particle displacement.1−8 From the practical standpoint, there has been recent discussion regarding the surprising importance of hydrodynamic interactions in crowded cellular environments.9−16 A common thread through these investigations is the importance of static and dynamic heterogeneity on the time scales of observation. In this article, we study a very simple model system, namely, a tracer colloidal particle immersed in a concentrated solution of larger spheres, and demonstrate that much of the phenomenology is captured in this system. There have been many recent examples of non-Gaussian behavior in diffusive systems.1−7 The quantity of interest is the self-part of the van Hove correlation function, Gs(r,t), which is the probability that a particle is at position r at time t given that it was at the origin at time t = 0. In the original development of Brownian motion, Gs(r,t) is Gaussian in r at long times when the mean-square displacement ⟨(Δr)2⟩ is linear in t. A number of experiments, for example, of vesicles in nematic fluids and colloids in a matrix of actin filaments,2,3 have shown that Gs(r,t) is exponential in r on time scales where ⟨(Δr)2⟩ ∼ t. This somewhat surprising result has been rationalized with the hypothesis that the dynamics of a single particle in a complex environment may be viewed as the superposition of several elementary diffusive processes with different “diffusion © XXXX American Chemical Society

constants”. A convolution of several Gaussian functions can result in apparent exponential behavior in Gs(r,t), while ⟨(Δr)2⟩ is always linear in t. A fundamental understanding of this behavior is of interest. As a caveat, we note that Gs(r,t) can be proven to be Gaussian only for a noninteracting Brownian particle diffusing in solvent. When interactions between particles are present Gaussian behavior is an approximation (Vineyard approximation) and not an exact result. The dynamics of proteins in crowded cellular environments plays an essential role in various cellular processes such as protein association reactions.11,17−20 One wonders if the density of the cytoplasm is optimized, via evolution, to maximize the rate of biochemical reactions. If the density of the cytoplasm were too low, chances of reactive proteins colliding with each other would be small. On the other hand, if the density of the cytoplasm were too high, the protein diffusion should slow down, which would hinder biochemical reactions.21−23 In such crowded cells, protein diffusion coefficients decrease by about an order of magnitude compared to the value in dilute solutions.24,25 Predicting the diffusion constant of proteins in cells is important in its own right because many biochemical reactions in cells are diffusioncontrolled. In addition to the non-Gaussian behavior discussed Special Issue: James L. Skinner Festschrift Received: February 1, 2014 Revised: April 1, 2014

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feasible for the time scales of interest. In implicit solvent models, HI are introduced through a resistance tensor. Stokesian dynamics9,50,51 approximates the resistance tensor by the sum of a many-body resistance tensor and a lubrication correction tensor. This method is regarded as the most advanced approach, but calculating the many-body resistance tensor is very computationally intensive. There are not many applications of Stokesian dynamics other than hard spheres. A notable exception is the study of the cytoplasm by Ando and Skolnick,9 who simulated a 20 component mixture model of the cytoplasm. In this work, we employ fast lubrication dynamics (FLD) in which the many-body resistance tensor is approximated by an isotropic matrix with short-range lubrication forces incorporated as in Stokesian dynamics, but with the far-field part treated in a mean-field fashion. This allows us to study the dynamics over long time scales. FLD has been employed successfully to investigate various colloidal suspensions including charged colloidal suspensions under shear.52,53 We study the dynamics of small spheres, at high dilution, in a suspension of large spheres. The simulations show that, even in this simple system, as the volume fraction of large spheres is increased, the Gs(r,t) of the small spheres shows non-Gaussian behavior at larger distances, under conditions where ⟨(Δr)2⟩ is linear in t. HI cause a significant slowing down of the particle dynamics. When the system is viewed at the same value of ⟨(Δr)2⟩, however, the qualitative behavior of Gs(r,t) with and without HI is similar, suggesting that the main effect of HI is to slow down the dynamics of the particles. The rest of this paper is organized as follows: In section II, we present simulation details; in section III, we present results; and in section IV, we present our conclusions.

above, there are many observations of anomalous subdiffusion,20,26−30 where ⟨(Δr)2⟩ ∼ tα with α < 1 in protein solutions. A recent simulation study suggests that hydrodynamic interactions play an important role in the crowded cytoplasm.9 There are three factors that could cause a reduction of the diffusion coefficient in crowded environments: excluded volume interactions, 31,32 nonspecific attractive interactions,33−36 and hydrodynamic interactions (HI).37−42 There have been extensive studies on excluded volume interactions and nonspecific interactions, but few studies have been carried on the effects of HI. Ando and Skolnick9 argued that HI were necessary to predict the reduction in the diffusion coefficient. However, understanding the effect of HI on protein diffusion in crowded environments still remains to be answered, partly because of the complexity of cells. There are therefore two broad issues of interest: What is the effect of local heterogeneity or separation of time scales on the dynamics of particles, and what is the effect of hydrodynamic interactions on the dynamics. In particular, do HI merely reduce the diffusion constant, or do they also have a qualitative effect on the distribution of single particle displacement? In this work, we address these issues using a simple model, namely, a binary mixture of Lennard-Jones particles with different collision diameters. The only difference is a size disparity between the two species. We study the effect of HI using the so-called fast lubrication dynamics approximation to the full Stokesian dynamics. To our knowledge, this is the first study of the effect of HI on a binary mixture. In crowded environments, diffusion becomes time-dependent. At short and long time scales, the colloids (or proteins) undergo Brownian motions with short and long time diffusion coefficients, respectively. The short time diffusion constant corresponds to the particle diffusing in pure solvent, while the long time diffusion constant corresponds to the particle diffusing in an effective medium composed of all the particles. At intermediate time scales, when the particles have not sufficiently sampled the heterogeneous solution, anomalous subdiffusive behavior,20,27 such as ⟨(Δr)2⟩ ∼ tα with α < 1, is often observed. This intermediate time scale may be divided even further into several regimes if systems are compartmentalized in a hierarchical fashion.28,43−45 Gs(r,t)) has been presumed to be a Gaussian function for Fickian dynamics when ⟨(Δr)2⟩ ∼ t. A recent study by Weiss30 showed clearly that even at an anomalous subdiffusive regime one may expect Gaussian dynamics. In colloidal solutions close to the glass transition, Hurtado et al.1 reported that Gs(r,t) was an exponential (not Gaussian) function of r at large r and argued that the exponential form could be a universal dynamic feature close to glass and jamming transitions. More interestingly, Wang et al.2,3 showed that Gs(r,t) of colloids in actin suspensions is exponential at large r even when ⟨(Δr)2⟩ ∼ t. We find similar behavior in our simple system of binary mixtures of colloids, suggesting it is a general feature of diffusion in solutions with dynamic disparities. In this paper, we study the dynamics of a binary mixture of spheres using Brownian dynamics with and without HI. Incorporating HI properly into molecular simulations of many-body systems has been a challenging task because it requires either a tremendous number of solvent molecules in explicit solvent models46−49 or highly intensive calculation of resistance tensors in implicit solvent models.50 For the systems of interest to us, the number of solvent molecules would be sufficiently large that explicit solvent simulations are not

II. MODEL AND SIMULATION METHODS The system consists of a binary mixture of colloids that interact via a Lennard-Jones potential. The collision diameter of small colloids is σ. We use σ as the unit of length in this work. The collision diameter (σl) of large colloids is 3 or 5. The interaction, VLJ, between spheres i and j is given by VLJ = 4ϵ[(σij/rij)12 − (σij/rij)6]), where rij is the distance between centers of the spheres, σij = (σi + σj)/2, and σi is the collision diameter of sphere i. We set ϵ = 4kBT, where kB and T denote the Boltzmann constant and temperature, respectively. The energy scale is defined by setting kBT = 1. The simulation cell is a cube of linear dimension L = 17.4 with Nl large particles and Ns small particles. Periodic boundary conditions are employed in all directions. The volume fractions of large and small colloids are defined as ϕs ≡ σ3l πNl/(6L3) and ϕs ≡ πNs/(6L3), respectively. The volume fraction of the small (tracer) colloids is set at ϕs = 0.002, and ϕl is varied from 0.2 to 0.6 (Figure 1). Initial configurations are generated by placing large and small colloids at random positions. Properties are averaged over 10 trajectories starting with different initial configurations. The system evolves via the Langevin equation m·

dv = fH + fB + fP dt

(1)

where m is the moment of inertia tensor, v is the velocity vector, and fH, fB, and fP denote force vectors due to HI, Brownian motion, and intermolecular interactions, respectively. When the inertial term is ignored, the above equation simplifies to B

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(BD) simulations are performed using the same algorithm by switching off the near-field lubrication forces (i.e., setting Rδ = 0). The following properties are calculated in the simulations. The mean-square displacement is defined as ⟨(Δr)2⟩ = ⟨|r(⃗ t) − r(⃗ 0)|2⟩, where r(⃗ t) is the position vector of a colloid at time t and ⟨···⟩ denotes an ensemble average. The self-part of the van Hove correlation function Gs(r,t) is given by Gs(r,t) = ⟨δ(r ⃗ − [r(⃗ t) − r(⃗ 0)])⟩, and its Fourier transform is the self-part of the ⃗ (⃗ t) − intermediate scattering function Fs(k,t) = ⟨exp{ik·(r r(⃗ 0))}⟩, where k⃗ is the momentum transfer variable vector and k ≡ |k|.⃗ In order to estimate Gs(r,t), we construct histograms with a bin size of 0.05. Because our systems are evolved up to t = 105 and histograms are estimated every t = 10 on the fly, the histograms are averaged over about 104 configurations. The histogram is normalized by the numbers of configurations and colloids to obtain Gs(r,t). The long time diffusion coefficients (DL) are obtained from the Einstein relation, DL = limt→∞⟨(Δr2)(t)⟩/6t, and the translational relaxation time (τα) is obtained from Fs(k = 2π/r,t = τα) = e−1. A recent study by Guan, Wang, and Granick8 reported nonGaussian dynamics of colloidal mixtures. In their study, the size of small colloids was 0.28 μm. In the limit without any large colloids, the diffusion coefficient of small colloids was about 0.6 μm2/s. If we should compare the size and diffusion coefficients to our simulation results, the relevant intrinsic time scale (or the unit time) in our simulations would be roughly 0.1 s.

Figure 1. Snapshot of the simulated system for ϕl = 0.6, ϕs = 0.002, and σl = 5.

fH + fB + fP = 0

(2)

In the limit of low Reynold number for colloid suspensions, fH can be expressed as a product of the resistance tensor R and the difference (v∞ − v) between the fluid velocity vector (v∞) and the particle velocity vector (v)54 (v∞ = 0 in our system). In FLD simulations, the resistance tensor R is divided into two parts

R = R 0 + Rδ

III. RESULTS AND DISCUSSION The dynamics of both the large and small colloids are characterized by three regimes, although the time scales are different. At short times, the particles undergo Brownian motion in solvent with a short time diffusion coefficient, DS. At long times, the particles show diffusive motion (with ⟨(Δr)2⟩ ∼ t) with a long time diffusion coefficient, DL. At intermediate times, there is a crossover between the short and long time behavior, and in this regime, the anomalous diffusion is observed with an apparent exponent α where ⟨(Δr)2⟩ ∼ tα. Figure 3a,b depicts ⟨(Δr)2⟩ as a function of time on a log− log plot for the large and small particles, respectively. The volume fraction (ϕs) of the small (tracer) colloids is set at ϕs = 0.002 in our study. The qualitative behavior is the same for the small and large particles. For a volume fraction of large particles of ϕl = 0.2, there is no intermediate regime, and DL and DS are almost the same. As the volume fraction of large particles is increased, the time before the long time regime is reached becomes longer, and for ϕl = 0.6, the long time regime is only reached for times on the order of t = 104. The asymptotic long time regime is reached at shorter times for the larger particles because the larger particles sample the environment of small particles at shorter length scales. The displacement of the small particles is larger, as expected. The qualitative features are similar to both BD and FLD simulations, although the anomalous regime appears at shorter times in BD simulations. Short-ranged lubrication force is incorporated in FLD simulations, which imposes an effective additional repulsion between colloids and slows down the dynamics of colloids. Therefore, it takes longer in FLD simulations for colloids to reach a subdiffusive anomalous regime than in BD simulations. The diffusion of both small and large particles is slowed down by hydrodynamic interactions. Figure 4 depicts (a) DL/ D0 for large colloids and (b) DL/Dϕl=0 for small colloids, where

(3)

where R0 and Rδ are isotropic tensors for the short time selfdiffusion (from random forces) and the near-field lubrication correction tensors, respectively.52,53,55 The Brownian forces, fB, are governed by fluctuation−dissipation theorem such that ⟨fB⟩ = 0 and ⟨fBfB⟩ = 2kBTR/Δt. fP is obtained from the LennardJones intermolecular interactions. More details on FLD simulations can be found elsewhere.52,53 The numerical method follows closely that proposed by Kumar and Higdon.52 We employ a midpoint algorithm developed by Banchio and Brady55 to evolve the system. Rδ is calculated with a cutoff distance rcut = 2.5 between two colloids. To avoid divergences in Rδ, we use a small time step of Δt = 0.0001. We test our FLD simulations by reproducing results for monodisperse colloidal suspensions obtained by Banchio and Brady.55 Figure 2 compares our simulations (for a one component colloidal dispersion) to theirs and shows that the results are within statistical uncertainties. Brownian dynamics

Figure 2. Comparison of our FLD simulation results for the short time diffusion coefficient of monodisperse colloids (triangles) to previous simulations of hard sphere colloids (circles).55 C

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diameter of large colloids is 5, this suggests that large colloids are virtually immobile at small and intermediate time scales. This suggests that the spatial heterogeneity might arise from the large colloids being essentially quenched over short time scales, as is the case in the Lorentz model20 where a particle diffuses in a sea of randomly arranged fixed obstacles. Unlike the Lorentz model, however, where matrix particles are immobile at all time scales, in our systems, the large colloids still diffuse at long time scales, which allows small colloids to sample various heterogeneous regions in simulation times. The decrease in the particle dynamics by HI is also manifested in an increase in the translational relaxation time τα, depicted in Figure 5. The increase in τα with volume fraction

Figure 3. Comparison of BD and FLD results for the mean-square displacement, ⟨(Δr)2⟩ for (a) large and (b) small particles, for various values of the volume fraction (ϕl) of large particles. The symbols in part (b) represent the time values used in Figure 7.

Figure 5. Comparison of BD and FLD results for the translation relaxation time, τα, as a function of ϕl for (a) large and (b) small particles.

is modest for ϕl ≤ 0.4 but increases sharply going to ϕl = 0.6. At the highest volume fraction, HI play a significant role in the slowing down of the single particle dynamics. The space-dependent diffusion of the particles is described by the self-correlation function Gs(r,t), depicted in Figure 6. For a single particle undergoing Brownian motion

Figure 4. Comparison of BD and FLD results for the relative long time diffusion coefficients (a) DL/D0 for large particles and (b) DL/ Dϕl=0 for small colloids as a function of ϕl. D0 and Dϕl=0 denote the diffusion coefficient of a large particle at infinite dilution and the diffusion coefficient of small particles with ϕl = 0 and ϕs = 0.002, respectively. Each point is labeled with the value of the relative long time diffusion coefficient.

Gs(r , t ) =

⎛ 1 ⎞3/2 ⎛ r 2 ⎞ ⎜ ⎟ exp⎜ − ⎟ ⎝ 4πDt ⎠ ⎝ 4Dt ⎠

(4)

where D is the diffusion constant. For the colloidal particles, we expect a Gaussian Gs(r,t) at short times corresponding to free diffusion in solvent. At long times, the particle position is uncorrelated with its starting point and we expect Gs(r,t) to be flat. The behavior at intermediate times can be interesting because the particles experience a heterogeneous environment over these time scales. The qualitative trends in Gs(r,t) are similar for the large and small colloidal particles. For the same value of time, the peak in 4πr 2 G s (r,t) occurs at small distances r when HI are incorporated. This is because the dynamics are slower and the particle has, on average, moved a shorter distance from the starting point. As t increases, the peak moves to larger distances, as expected. The peak in 4πr2Gs(r,t) is also at shorter distances for the large particles (when compared to the small particles) because they have a lower diffusion constant.

D0 is the diffusion coefficient of a large colloid at infinite dilution and Dϕl=0 is the diffusion coefficient of small colloids for ϕl = 0 and ϕs = 0.002. HI decrease the value of DL/D0 by approximately the same percentage for the large and small particles, and the percentage decrease increases with increasing volume fraction, from about 20−30% for ϕl = 0.2 to over 80% for ϕl = 0.6. The significant decrease in the long time diffusion constant for crowded systems is consistent with previous simulations of a model cytoplam.9 The diffusion of large colloids is much slower than that of small colloids, for example, when ⟨(Δr)2⟩ ∼ 1 for small colloids and ⟨(Δr)2⟩ ∼ 0.1 for large colloids. Considering that the D

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r.2,3,6 As depicted in Figure 7C, Gs(r,t) of large colloids are Gaussian at t = 1000 when ⟨(Δr)2⟩ ∼ t for large colloids. We also compare the time-averaged Gs(r,t) of each small colloid to the ensemble-averaged Gs(r,t). We find that timeaveraged Gs(r,t) values are almost identical to the ensembleaveraged one; that is, our systems are ergodic. Previous studies suggested that the exponential Gs(r,t) could be interpreted as a linear combination of Gaussian Gs(r,t) values with different diffusion coefficients. In our colloid mixtures, even if each single colloid would diffuse with a different transient diffusion coefficient in a local region, the single colloid would sample the whole system, and Gs(r,t) of a single colloid should be identical to the ensemble-averaged quantity. The non-Gaussian Gs(r,t) can be also observed in mixtures of colloids with a different smaller size ratio. We perform BD simulations and analysis for a mixture of large and small particles with a different size ratio of 3. Note that the volume fraction of small particles is identical to 0.002 as in the above simulations. The long time diffusion coefficient (DL) for small particles is DL = 0.00085 for ϕl = 0.6. Figure 8 depicts both BD

Figure 6. Comparison of BD and FLD results for the self-part of the van Hove correlation function, plotted as 4πr2Gs(r,t) for (a) large and (b) small particles for various times and for ϕl = 0.6.

At low volume fractions, Gs(r,t) is a Gaussian function of r at all times, but at the highest volume, Gs(r,t) of small colloids is exponential for large distances. We also investigate a graph of ln(Gs(r,t)/A)/r as a function of r to check whether Gs(r,t) of small colloids is exponential at large r. Here, A is the parameter in the fitting function A exp(−r/B). We fit simulation results of Gs(r,t) to the function and determined A. We found that ln(Gs(r,t)/A)/r reached a plateau at large r, corroborating that Gs(r,t) was exponential. At low volume fractions, BD and FLD results are both fit with Gaussians with different values of the diffusion constant. At high volume fraction, Gs(r,t) of small colloids shows two regimes in r at large times. Figure 7A,B depicts Gs(r,t) as a function of r for the small particles, for ϕl = 0.6, and various times. The times, t*, are chosen so that the mean-square displacement has the values noted in Figure 3. The three time points chosen are depicted with symbols in Figure 3: all lie in the long time regime where the particles are diffusive. In all the cases shown, Gs(r,t) is Gaussian at short distance and exponential at larger distances, at scales corresponding to 6 ≤ ⟨(Δr)2⟩ ≤ 150. The qualitative features are similar in the BD and FLD simulations, and the curves for the two models overlap in Figure 7 when Gs(r,t) values are plotted at times when the ⟨(Δr)2⟩ is the same in the two models. The simulation results are consistent with recent experiments by Wang et al., where colloid beads on phospholipid bilayer tubes or in entangled actin suspensions enter Fickian regimes while Gs(r,t) was still exponential at large

Figure 8. Simulation results for Gs(r,t) of small particles for ϕl = 0.6 and the size ratio of 3. Solid lines are theoretical predictions using eq 4 and the long time diffusion coefficient of 0.0085.

simulation results (symbols) and theoretical predictions (lines) of Gs(r,t) for small particles at three different times. Theoretical predictions are made using eq 4 and DL = 0.0085. Similarly to the case of size ratio of 5, BD simulations show a non-Gaussian Gs(r,t) at relatively large r, which deviates from Gaussian theoretical predictions for all three times.

IV. SUMMARY AND CONCLUSIONS We investigate the dynamics of a binary mixture of colloidal particles using Brownian dynamics with and without hydrodynamic interactions. The two components of the mixture differ in size with a size ratio of 5. Both the large and small

Figure 7. (A) BD and FLD simulation results for Gs(r,t) of small particles for ϕl = 0.6. The time t* is chosen so that the mean-square displacement has the values shown in the legend. Solid and dashed lines are fits to simulation results using Gaussian and exponential functions, respectively. FLD simulation results for Gs(r,t) for (B) small and (C) large particles for ϕl = 0.6 for various times. E

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(5) He, K.; Khorasani, F. B.; Retterer, S. T.; Thomas, D. K.; Conrad, J. C.; Krishnamoorti, R. Diffusive Dynamics of Nanoparticles in Arrays of Nanoposts. ACS Nano 2013, 7, 5122−5130. (6) Kim, J.; Kim, C.; Sung, B. J. Simulation Study of Seemingly Fickian but Heterogeneous Dynamics of Two Dimensional Colloids. Phys. Rev. Lett. 2013, 110, 047801. (7) Bhattacharya, S.; Sharma, D. K.; Saurabh, S.; De, S.; Sain, A.; Nandi, A.; Chowdhury, A. Plasticization of Poly(vinylpyrrolidone) Thin Films under Ambient Humidity: Insight from Single-Molecule Tracer Diffusion Dynamics. J. Phys. Chem. B 2013, 117, 7771−7782. (8) Guan, J.; B, W.; Granick, S. Even Hard-Sphere Colloidal Suspensions Display Fickian Yet Non-Gaussian Diffusion. ACS Nano 2014, 8, 3331−3336. (9) Ando, T.; Skolnick, J. Crowding and Hydrodynamic Interactions Likely Dominate in Vivo Macromolecular Motion. Proc. Natl. Acad. Sci. U.S.A. 2010, 107, 18457−18462. (10) 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. (11) Dill, K. A.; Ghosh, K.; Schmit, J. D. Physical Limits of Cells and Proteomes. Proc. Natl. Acad. Sci. U.S.A. 2011, 108, 17876−17882. (12) McGuffee, S. R.; Elcock, A. H. Diffusion, Crowding & Protein Stability in a Dynamic Molecular Model of the Bacterial Cytoplasm. PLoS Comp. Biol. 2010, 6, e1000694. (13) Dix, J. A.; Verkman, A. S. Crowding Effects on Diffusion in Solutions and Cells. Annu. Rev. Biophys. 2008, 37, 247−263. (14) Roman, S. G.; Chebotareva, N. A.; Eronina, T. B.; Kleymenov, S. Y.; Makeeva, V. F.; Poliansky, N. B.; Muranov, K. O.; Kurganov, B. I. Does the Crowded Cell-like Environment Reduce the Chaperone-like Activity of α-Crystallin? Biochemistry 2011, 50, 10607−10623. (15) Niwayama, R.; Shinohara, K.; Kimura, A. K. Hydrodynamic Property of the Cytoplasm Is Sufficient To Mediate Cytoplasmic Streaming in the Caenorhabiditis elegans Embryo. Proc. Natl. Acad. Sci. U.S.A. 2011, 108, 11900−11905. (16) Zhou, H.-X.; Rivas, G.; Minton, A. P. Macromolecular Crowding and Confinement: Biochemical, Biophysical, and Potential Physiological Consequences. Annu. Rev. Biophys. 2008, 37, 375−397. (17) Frembgen-Kesner, T.; Elcock, A. H. Absolute Protein−Protein Association Rate Constants from Flexible, Coarse-Grained Brownian Dynamics Simulations: The Role of Intermolecular Hydrodynamic Interactions in Barnase-Barstar Association. Biophys. J. 2010, 99, L75− L77. (18) Kim, J. S.; Yethiraj, A. Crowding Effects on Association Reactions at Membranes. Biophys. J. 2010, 98, 951−958. (19) Kim, J. S.; Yethiraj, A. Crowding Effects on Protein Association: Effect of Interactions between Crowding Agents. J. Phys. Chem. B 2011, 115, 347−353. (20) Höefling, F.; Franosch, T. Anomalous Transport in the Crowded World of Biological Cells. Rep. Prog. Phys. 2013, 76, 046602. (21) Gershon, N. D.; Porter, K. R.; Trus, B. L. The Cytoplasmic Matrix: Its Volume and Surface Area and the Diffusion of Molecules through It. Proc. Natl. Acad. Sci. U.S.A. 1985, 82, 5030−5034. (22) Minton, A. P. Confinement as a Determinant of Macromolecular Structure and Reactivity. Biophys. J. 1992, 63, 1090−1100. (23) Ellis, R. J. Macromolecular Crowding: Obvious but under Appreciated. Trends. Biochem. Sci. 2001, 26, 597−604. (24) Muramatsu, N.; Minton, A. P. Tracer Diffusion of Globular Proteins in Concentrated Protein Solutions. Proc. Natl. Acad. Sci. U.S.A. 1988, 85, 2984−2988. (25) Konopka, M. C.; Shkel, I. A.; Cayley, S.; Record, M. T.; Weisshaar, J. C. Crowding and Confinement Effects on Protein Diffusion in Vivo. J. Bacteriol. 2006, 188, 6115−6123. (26) Szymanski, J.; Weiss, M. Elucidating the Origin of Anomalous Diffusion in Crowded Fluids. Phys. Rev. Lett. 2009, 103, 038102. (27) Bouchaud, J. P.; Georges, A. Anomalous Diffusion in Disordered Media: Statistical Mechanisms, Models and Physical Applications. Phys. Rep. 1990, 195, 127−293.

particles display three dynamical regimes: At short times and long times, the mean-square displacement is linear in time. These correspond to the particles diffusing in solvent and in an effective medium composed of all the particles. At intermediate times, subdiffusion is observed as the dynamics transitions between the two limiting cases. Hydrodynamic interactions result in a significant slowing down of the dynamics, even at the highest volume fraction, consistent with previous simulations of a model cytoplasm. The qualitative features of the dynamical behavior, however, are not sensitive to the incorporation of HI. The three time regimes described above are similar to and without HI, although the onset of long time behavior occurs at longer times with HI, consistent with a lower diffusion constant. The self-correlation function is insensitive to HI as long as the BD and FLD results are compared at times where the mean-square displacement is the same. The simulations shed light on the experimental observations of a non-Gaussian self-correlation function under conditions where the mean-square displacement is linear in time. At both short and very long times, Gs(r,t) is a Gaussian function of r, but this is not the case in the intermediate regime where it is Gaussian at small r but exponential at large r. There is a regime of time, however, where the mean-square displacement is linear in time but Gs(r,t) is non-Gaussian, similar to what is seen in the experiments. We interpret this as a case where the dynamics are still in the crossover regime between short time and long time behavior, and the time for the self-correlation function to reach the asymptotic behavior is longer than the time for the mean-square displacement to become linear in time. The simulations demonstrate that hydrodynamic interactions are important in colloidal mixtures, but the effect is of a quantitative (although large) but not qualitative nature. The simulations also provide an explanation of the “diffusive but not Gaussian” behavior seen in experiments.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Research Foundation of Korea (NRF) under Grant No. NRF-2011-220-C00030. This research was also supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2013R1A1A2009972).

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REFERENCES

(1) Hurtado, P. I.; Berthier, L.; Kob, W. Heterogeneous Diffusion in a Reversible Gel. Phys. Rev. Lett. 2007, 98, 135503. (2) Wang, B.; Anthony, S. M.; Bae, S. C.; Granick, S. Anomalous yet Brownian. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 15160−15164. (3) Wang, B.; Kuo, J.; Bae, S. C.; Granick, S. When Brownian Diffusion Is Not Gaussian. Nat. Mater. 2012, 11, 481−485. (4) Skaug, M. J.; Mabry, J.; Schwartz, D. K. Intermittent Molecular Hopping at the Solid−Liquid Interface. Phys. Rev. Lett. 2013, 110, 256101. F

dx.doi.org/10.1021/jp5011617 | J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

Article

(28) Kusumi, A.; Nakada, C.; Ritchie, K.; Murase, K.; Suzuki, K.; Murakoshi, H.; Kasai, R. S.; Kondo, J.; Fujiwara, T. Paradigm Shift of the Plasma Membrane Concept from the Two-Dimensional Continuum Fluid to the Partitioned Fluid: High-Speed SingleMolecule Tracking of Membrane Molecules. Annu. Rev. Biophys. Biomol. Struct. 2005, 34, 351−354. (29) Wong, I. Y.; Gardel, M. L.; Reichman, D. R.; Weeks, E. R.; Valentine, M. T.; Bausch, A. R.; Weitz, D. A. Anomalous Diffusion Probes Microstructure Dynamics of Entangled F-Actin Networks. Phys. Rev. Lett. 2004, 92, 178101. (30) Weiss, M. Single-Particle Tracking Data Reveal Anticorrelated Fractional Brownian Motion in Crowded Fluids. Phys. Rev. E 2013, 88, 010101(R). (31) Dix, J. A.; Verkman, A. S. Crowding Effects on Diffusion in Solutions and Cells. Annu. Rev. Biophys. 2008, 37, 247−263. (32) Zhou, H. X.; Rivas, G.; Minton, A. P. Macromolecular Crowding and Confinement: Biochemical, Biophysical and Potential Physiological Consequences. Annu. Rev. Biophys. 2008, 37, 375−397. (33) Wang, Y.; Li, C.; Pielak, G. J. Effects of Proteins on Protein Diffusion. J. Am. Chem. Soc. 2010, 132, 9392−9397. (34) Elcock, A. H. Models of Macromolecular Crowding Effects and the Need for Quantitative Comparisons with Experiment. Curr. Opin. Struct. Biol. 2010, 20, 196−206. (35) Rosen, J.; Kim, Y. C.; Mittal, J. Modest Protein-Crowder Attractive Interactions Can Counteract Enhancement of Protein Association by Intermolecular Excluded Volume Interactions. J. Phys. Chem. B 2011, 115, 2683−2689. (36) Geyer, T. Mixing Normal and Anomalous Diffusion. J. Chem. Phys. 2012, 137, 115101. (37) Bussell, S. J.; Koch, D. L.; Hammer, D. A. Effect of Hydrodynamic Interactions on the Diffusion of Integral Membrane Proteins: Tracer Diffusion in Organelle and Reconstituted Membranes. Biophys. J. 1995, 68, 1828−1835. (38) Furukawa, A.; Tanaka, H. Key Role of Hydrodynamic Interactions in Colloidal Gelation. Phys. Rev. Lett. 2010, 104, 245702. (39) Frembgen-Kesner, T.; Elcock, A. H. Striking Effects of Hydrodynamic Interactions on the Simulated Diffusion and Folding of Proteins. J. Chem. Theory Comput. 2009, 5, 242−256. (40) Rogers, S. A.; Lisicki, M.; Cichocki, B.; Dhont, J. K. G.; Lang, P. R. Rotational Diffusion of Spherical Colloids Close to a Wall. Phys. Rev. Lett. 2012, 109, 098305. (41) Michailidou, V. N.; Petekidis, G.; Swan, J. W.; Brady, J. F. Dynamics of Concentrated Hard-Sphere Colloids near a Wall. Phys. Rev. Lett. 2009, 102, 068302. (42) Bernadó, P.; de la Torre, J. G.; Pons, M. Macromolecular Crowding in Biological Systems: Hydrodynamics and NMR Methods. J. Mol. Recognit. 2004, 17, 397−407. (43) Ritchie, K.; Lino, R.; Fujiwara, T.; Murase, K.; Kusumi, A. The Fence and Picket Structure of the Plasma Membrane of Live Cells As Revealed by Single Molecule Techniques. Mol. Membr. Biol. 2003, 20, 13−18. (44) Suzuki, K.; Ritchie, K.; Kajikawa, E.; Fujiwara, T.; Kusumi, A. Rapid Hop Diffusion of a G-Protein-Coupled Receptor in the Plasma Membrane as Revealed by Single-Molecule Techniques. Biophys. J. 2005, 88, 3659−3680. (45) Sung, B. J.; Yethiraj, A. Computer Simulations of Protein Diffusion in Compartmentalized Cell Membranes. Biophys. J. 2009, 97, 472−479. (46) Takemura, K.; Kitao, A. Effects of Water Model and Simulation Box Size on Protein Diffusional Motions. J. Phys. Chem. B 2007, 111, 11870−11872. (47) Harada, R.; Sugita, Y.; Feig, M. Protein Crowding Affects Hydration Structure and Dynamics. J. Am. Chem. Soc. 2012, 134, 4842−4849. (48) Morrone, J. A.; Li, J.; Berne, B. J. Interplay between Hydrodynamics and the Free Energy Surface in the Assembly of Nanoscale Hydrophobes. J. Phys. Chem. B 2012, 116, 378−389.

(49) Li, J.; Morrone, J. A.; Berne, B. J. Are Hydrodynamic Interactions Important in the Kinetics of Hydrophobic Collapse? J. Phys. Chem. B 2012, 116, 11537−11544. (50) Durlofsky, L.; Brady, J. F.; Bossis, G. Dynamic Simulation of Hydrodynamically Interacting Particles. J. Fluid. Mech. 1987, 180, 21− 49. (51) Brady, J. F.; Bossis, G. Stokesian Dynamics. Annu. Rev. Fluid. Mech. 1988, 20, 111−157. (52) Kumar, A.; Higdon, J. J. L. Origins of the Anomalous Stress Behavior in Charged Colloidal Suspensions under Shear. Phys. Rev. E 2010, 82, 051401. (53) Bybee, M. D. Hydrodynamic Simulations of Colloidal Gels: Microstructure, Dynamics, and Rheology. Ph.D. Thesis, University of Illinois at UrbanaChampaign, 2009. (54) Sierou, A.; Brady, J. F. Accelerated Stokesian Dynamics Simulations. J. Fluid. Mech. 2001, 448, 115−146. (55) Banchio, A. J.; Brady, J. F. Accelerated Stokesian Dynamics: Brownian Motion. J. Chem. Phys. 2003, 118, 10323−10332.

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