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# Dynamic cluster formation determines viscosity and diffusion in dense protein solutions

Edited by Huan-Xiang Zhou, University of Illinois at Chicago, Chicago, IL, and accepted by Editorial Board Member J. A. McCammon April 1, 2019 (received for review October 11, 2018)

## Significance

For living cells to function, proteins must efficiently navigate the densely packed cytosol. Protein diffusion is slowed down by high viscosity and can come to a complete halt because of nonspecific binding and aggregation. Using molecular dynamics simulations, we develop a detailed description of protein diffusion in concentrated protein solution. We confirm that soluble proteins in concentrated solutions diffuse not as isolated particles, but as members of transient clusters between which they constantly exchange. Nonspecific protein binding and the formation of dynamic clusters nearly quantitatively account for the high viscosity and slow diffusivity in concentrated protein solutions, consistent with the Stokes–Einstein relations.

## Abstract

We develop a detailed description of protein translational and rotational diffusion in concentrated solution on the basis of all-atom molecular dynamics simulations in explicit solvent. Our systems contain up to 540 fully flexible proteins with 3.6 million atoms. In concentrated protein solutions (100 mg/mL and higher), the proteins ubiquitin and lysozyme, as well as the protein domains third IgG-binding domain of protein G and villin headpiece, diffuse not as isolated particles, but as members of transient clusters between which they constantly exchange. A dynamic cluster model nearly quantitatively explains the increase in viscosity and the decrease in protein diffusivity with protein volume fraction, which both exceed the predictions from widely used colloid models. The Stokes–Einstein relations for translational and rotational diffusion remain valid, but the effective hydrodynamic radius grows linearly with protein volume fraction. This increase follows the observed increase in cluster size and explains the more dramatic slowdown of protein rotation compared with translation. Baxter’s sticky-sphere model of colloidal suspensions captures the concentration dependence of cluster size, viscosity, and rotational and translational diffusion. The consistency between simulations and experiments for a diverse set of soluble globular proteins indicates that the cluster model applies broadly to concentrated protein solutions, with equilibrium dissociation constants for nonspecific protein–protein binding in the K*d* ≈ 10-mM regime.

The interior of cells is a densely crowded medium, in which macromolecular concentrations range from 90 mg/mL in red blood cells to 300 mg/mL in the mitochondrial matrix (1, 2). Macromolecular crowding influences the stability of proteins, reaction rates, the catalytic activity of enzymes, protein–protein association, and diffusion (3⇓⇓⇓⇓⇓⇓⇓⇓⇓–13). Excluded volume through steric repulsion (14) and attractive protein–protein interactions as well as hydrodynamic interactions affect protein diffusion (6, 15⇓⇓⇓–19). To address the influence of specific protein–protein interactions on protein diffusion (20), crowded solutions with proteins serving as both agents and readout have been studied (5, 14, 16, 21⇓⇓⇓⇓⇓⇓⇓⇓–30).

Experimental techniques to study the effects of macromolecular crowding on diffusion (15) include tracer boundary spreading (14), light scattering spectroscopy (31), fluorescence recovery after photobleaching (FRAP) (32⇓–34), electron spin resonance (35), single-particle tracking (36), fluorescence correlation spectroscopy (FCS) (37⇓–39), quasielastic neutron backscattering (27, 40), and NMR spectroscopy (24, 41, 42). Particle-based simulations complement these experiments (15), treating the proteins as spheres or ellipsoids (20, 43, 44), with residue-level coarse graining (45⇓–47), or as rigid all-atom models (16, 48). Hydrodynamic interactions contribute significantly to the slowdown of protein diffusion in crowded environments (19). In implicit solvent, they are ignored or approximated via the diffusion tensor (16, 19, 44, 49).

Rapid advances in computing hardware and simulation algorithms have opened up the opportunity to study macromolecular crowding using atomistic molecular dynamics (MD) simulations. Explicit solvent accounts directly for excluded volume effects and hydrodynamic interactions and mediates short-range attractive and long-range electrostatic protein–protein interactions (5, 28⇓–30, 50⇓–52). Here, we use atomistic MD simulations of dense protein solutions to calculate the viscosity and protein diffusion coefficients as a function of protein concentration (Fig. 1). Ubiquitin (UBQ), the third IgG-binding domain of protein G (GB3), hen egg white lysozyme (LYZ), and villin headpiece (VIL) are used as model proteins.

Soluble proteins self-associate in concentrated solution to form transient and dynamic clusters (19, 24, 53⇓⇓⇓⇓–58). Clustering has also been reported for membrane proteins (59). The influence of cluster formation on the protein translational and rotational diffusivity has recently been addressed by atomistic simulation studies (29, 30). Here, we build on these findings and put cluster formation in the framework of the Stokes–Einstein relations connecting viscosity, cluster size, and diffusion. Central questions are (*i*) whether the Stokes–Einstein relations remain valid in concentrated protein solutions (60), (*ii*) how transient protein interactions affect the diffusivity and apparent hydrodynamic radii of proteins in concentrated solutions (42), (*iii*) how viscosity depends on protein concentration (39, 61, 62), and (*iv*) whether colloid models apply to concentrated protein solutions.

To address these questions, we perform extensive MD simulations, develop a cluster model of concentration-dependent protein diffusion, and compare our results to the theoretical predictions for hard-sphere (HS) colloidal suspensions without and with attractive interactions. We put our findings in the context of a wide range of experimental and simulation studies and obtain a remarkably consistent picture of the diffusive dynamics in concentrated solutions of soluble globular proteins.

## Results

### Shear Viscosity Increases Strongly with Protein Volume Fraction.

The viscosity of concentrated protein solutions and of TIP4P-D solvent at different ion concentrations was calculated from the pressure tensor fluctuations (*SI Appendix*, Fig. S5). The quadratic function Eq. **3** fits

The viscosity of dense protein solutions exceeds the Einstein prediction **3** for the viscosity is related to the attraction strength, as measured by the osmotic virial coefficient (67⇓–69). In the following, we use **3** with values of b listed in Table 1 to account for the dependence of the viscosity on the protein volume fraction.

### Translational Diffusion Slows Down at High Protein Density.

As shown in Movie S1 for GB3 at 200 mg/mL with *SI Appendix*, Eq. **S5** (*SI Appendix*, Fig. S6). The MSD curves of the dense protein solutions averaged over starting times and proteins are linear at times exceeding 10 ns. The translational diffusion coefficients **7**, where we used **3**. The values before finite-size correction are listed in *SI Appendix*, Table S3.

After finite-size correction, the translational diffusion coefficient *A*). The translational diffusion coefficients calculated for the large systems (with **8**, developed below, accounts nearly quantitatively for the slowdown of translational diffusion with increasing concentration.

### Crowding Strongly Affects Rotational Diffusion.

Rotational diffusion coefficients *SI Appendix*). Fits to elements of the quaternion covariance matrix are shown in *SI Appendix*, Fig. S7. The resulting rotational diffusion coefficients *SI Appendix*, Fig. S8. Fits to the orientational correlation function are shown in *SI Appendix*, Fig. S9.

Orientationally averaged diffusion coefficients *B*). At infinite dilution, the UBQ and GB3 results are bracketed by the rotational diffusion coefficients obtained from Hydropro (70) calculations and from NMR spectroscopy (77, 78). The experimental rotational diffusion coefficient in dilute LYZ solution reported in ref. 73 is slightly lower than the calculated values, whereas the rotational diffusion coefficients reported in ref. 25 agree well with our data at all protein concentrations. The calculated rotational diffusion coefficients of dilute LYZ and VIL are in fair agreement with Hydropro (70) predictions. As for the translational diffusion, the rotational diffusion coefficients calculated for the large systems (**9** predicts the rotational diffusion coefficients of UBQ, GB3, VIL, and LYZ accurately over the entire concentration range, except for the LYZ solution at 100 mg/mL concentration, where the effect of the weak clustering (Fig. 3*C*) is somewhat overestimated.

### Diffusion in Dense Protein Solutions Follows the Stokes–Einstein Relation.

Given translational and rotational diffusion coefficients, the viscosity can be estimated from the Stokes–Einstein relations (Eq. **6**). For all small systems (**6**, because the results are quite sensitive to the uncertainties in

### Hydrodynamic Radius, Cluster Size, and Diffusion Are Related.

We obtained very similar hydrodynamic radii from the Stokes–Einstein relations for translation and rotation, Eqs. **10** and **11**, respectively (Fig. 3*D*). Therefore, after correcting for finite-size effects with actual shear viscosities

If the increase in viscosity were to capture all factors that contribute to the concentration-dependent slowdown of protein diffusivity, then the hydrodynamic radius, calculated from the Stokes–Einstein relations, Eqs. **10** and **11**, should remain constant at all concentrations. Instead, we observe that the effective hydrodynamic radius cubed, *C*). Indeed, when calculating the cluster size distribution based on an α-carbon distance cutoff criterion, the cluster size distribution shifts to larger clusters at increasing protein volume fraction (*SI Appendix*, Fig. S10). For protein concentrations up to 100 mg/mL, the mean number of proteins in a cluster grows linearly as *C*). The highest concentration (200 mg/mL) was not included in the fit, because the close proximity of proteins causes a significant dependence of the calculated mean cluster size on the cutoff criterion (*SI Appendix*, Fig. S11). Given the linear increase of cluster size with protein volume fraction ϕ, the effective hydrodynamic radius cubed should likewise increase linearly with ϕ, *D*).

As shown in *SI Appendix*, the clustering propensity ζ is related to an effective dissociation constant *SI Appendix*, Fig. S12), we obtain dissociation constants of

### Effective Viscosity Accounts for Hydrodynamic Interactions.

We investigated whether the effective viscosity *SI Appendix*, Fig. S13). For these trajectory segments, we calculated MSD curves (*SI Appendix*, Fig. S14) and translational diffusion coefficients *SI Appendix*, Fig. S15). We conclude that the effective viscosity indeed accounts for the hydrodynamic contributions to the diffusivity slowdown.

### Displacement Pair Correlation Shows Contribution from Direct and Hydrodynamic Interactions.

We calculated the displacement pair correlation introduced by Ando and Skolnick (19) (*SI Appendix*). We analyzed the protein pair correlation for pairs at distances 0.6–3 nm. At distances corresponding to cluster formation, we observed highly correlated motion for all protein pairs at all concentrations and time delays (*SI Appendix*, Fig. S16). At increasing pair distance (∼2–3 nm), the pair correlation decreased gradually.

### Protein Binding Interfaces.

The interactions between the proteins in clusters were loose but not entirely random in their orientation (*SI Appendix*, Fig. S17). For UBQ, the preferred binding interface coincides remarkably well with the noncovalent dimer interface reported from NMR measurements (82), more or less independent of protein concentration (Fig. 4). It includes residues 8–11, 20, 24, 25, 28, 31–42, 46–49, 54–60, and 71–76. The C-terminal tail (residues 71–76) and an adjacent relatively hydrophobic surface patch show strong involvement, in line with experimental evidence (82). In dense LYZ solutions, we observed that residues Asp48 and Arg73 contribute most to LYZ–LYZ interaction (*SI Appendix*, Fig. S17). In a Brownian dynamics study (83), these residues were found to play crucial roles in the formation of a LYZ–LYZ encounter complex.

### Colloidal Suspension Model.

Baxter’s attractive (sticky) HSs (68, 69, 84⇓–86) are widely used as a model for suspensions of interacting colloidal particles. Their association constant is related to the dimensionless Baxter parameter τ as *SI Appendix*), where *C*). Considering the rough approximations of this model, the cluster sizes of MC simulations of sticky HSs are in surprisingly good agreement with those of the atomistic MD simulations. We can also relate τ and *SI Appendix*, Fig. S18) (no data for LYZ and VIL, as we did not conduct simulations at low concentrations for these proteins). With

For the sticky HS model, Cichocki and Felderhof (68) derived a low-density expansion of the long-time single-particle diffusion coefficient in terms of the volume fraction,*A*, dashed-dotted line) from a fit to our protein simulation data for UBQ and GB3 at low ϕ, which gives us a value of *A*), we also calculated

Cichocki and Felderhof (68) also evaluated the quadratic term in the viscosity expansion,**3** and Table 1), we obtain values of

### Dissociation Constant from Off Rate of Nonspecific Complexes.

The cumulative distribution functions of the lifetimes of protein pairs (*SI Appendix*, Fig. S19) show that most pairs stay together for 1–50 ns, indicating dynamic clustering according to Liu et al.’s (24) terminology. The lifetimes of protein pairs are independent of the protein concentration, supporting the presence of dynamic protein clusters rather than protein aggregation. From the cumulative distribution function, we obtained the same median protein pair lifetime of ^{−1}, assuming exponential kinetics. Assuming in addition a Smoluchowski on rate, ^{−1}. The resulting “kinetic” dissociation constant

## Discussion

### Relative Slowdown in Diffusion Is Consistent with Experiment and Cluster Model.

The relative slowdown of translational and rotational diffusion of UBQ, GB3, LYZ, and VIL at increasing protein volume fraction is within the range of published experimental and simulation results. Literature data on the protein-concentration dependence of

The dependence of *A* and *SI Appendix*, Fig. S20*A*), considering that they cover both experiments and simulations and report results for proteins of different size, shape, and charge, for different model resolution and different experimental conditions (temperature, pH value). In particular, our simulation results for *A* and open green symbols in *SI Appendix*, Fig. S20*A*). We conclude that the slowdown of translational diffusion in concentrated protein solutions is a general feature of soluble proteins and that the extent of the slowdown is governed by the clustering propensity of the protein (notwithstanding additional shape effects, which may not be captured by our choice of globular proteins). We also conclude that the colloidal models of noninteracting HSs (94, 95) (*SI Appendix* and dashed and dotted lines in Fig. 5) significantly underestimate the slowdown of translational diffusion of most proteins studied.

We collected data on the slowdown of the rotational diffusion from experimental and simulation studies and normalized the diffusion coefficients to *B* and *SI Appendix*, Fig. S20*B*) than the slowdown of *C* and ζ in Table 1). Indeed, the slowdown of *SI Appendix*, Fig. S10). From small-angle X-ray scattering data, Stradner et al. (79) inferred slightly larger LYZ clusters (green crosses in Fig. 3*C*) compared with our simulations. However, the higher pH in their experiments may have reduced the net protein charge and increased cluster formation. Scattering data were subsequently found to be compatible with lysozyme being largely repulsive (98).

We stress that our dynamic cluster model (Eqs. **8** and **9** and solid lines in Figs. 3 *A* and *B* and 5) predicts the relative slowdown in diffusion based on protein cluster size and viscosity. In light of this, the prediction of the model represents the observed and calculated slowdown in diffusion strikingly well. Our cluster model relates differences in the relative diffusivities of different proteins to differences in their nonspecific interactions. This observation provides a physical basis for the protein-dependent concentration scaling factor ξ introduced by ref. 21 to establish consistency with a HS diffusion model (95).

### Separating the Effects of Clustering and Hydrodynamics.

Using an elegant MD simulation setup, Nawrocki et al. (30) found that direct protein interactions are the dominant contributors to the slowdown of rotational diffusivity, whereas hydrodynamics play only a minor role. Here, we could show that the effective viscosity accounts for the indirect, hydrodynamic effects of dense solutions on the diffusivity slowdown (*SI Appendix*, Fig. S15). In dense UBQ, GB3, and VIL solutions at 200 mg/mL, the viscosity is ∼2.5-fold increased compared to the solvent viscosity (Fig. 2). The translational diffusion decreases by a factor of ∼4, whereas the rotational diffusion decreases by a factor of ∼6 (Fig. 3 *A* and *B*). The direct effect of protein clustering (i.e., the increase in the effective hydrodynamic radius) accounts for an additional factor ∼

The displacement pair correlation function assesses concerted protein motion and is used to distinguish short-range and long-range interactions in simulations of crowded systems (19). At short distances, the proteins form clusters, and their motion is highly correlated (*SI Appendix*, Fig. S16). The pair correlation decreases at larger protein pair distances, because the short-range interactions fade out and only contributions from long-range effects (electrostatics, hydrodynamics) remain. Interestingly, at low protein concentrations (30–50 mg/mL), the protein motions are correlated up to high distances, whereas the correlation decreases quickly with increasing distance at high protein concentration. This suggests that the motion of proteins at low concentration is more effectively coupled by hydrodynamic and electrostatic forces. At high concentrations, the coupled motion of a protein pair at short distances appears to be effectively quenched by interference of competing proteins. This trend is surprising, because proteins may serve as bridging intermediates at high concentrations, which should effectively increase the pair correlation for the distances considered.

### Protein Solutions as Colloidal Suspensions with Attractions.

The effective dissociation constants

The formation of 3D clusters explains the strong increase of the mean cluster size *C*). Sticky spheres have no orientational preference, which allows the formation of compact clusters. By contrast, protein–protein interactions are directional and only some protein orientations result in favorable interactions (*SI Appendix*, Fig. S17). The orientational preference disfavors the formation of compact clusters, as can be seen in the formation of only *SI Appendix*, Fig. S21). In consideration of these differences, τ should be considered an effective parameter whose value depends on the property that is fitted and on the fit range. Therefore, we emphasize that the sticky HS model is only a rough approximation of a complex protein solution, which nonetheless explains the behavior of the complex protein systems surprisingly well up to intermediate protein concentration.

From light-scattering experiments, Scherer et al. (57) estimated

### Limitations.

The generality of our findings on the connection of cluster size, viscosity, and diffusion is subject to several limitations.

All four proteins studied here are small and globular. This justifies the use of averaged rotational diffusion coefficients

Slight finite-size effects on protein clustering and diffusivity were observed. In the smallest simulation of dense protein solutions (UBQ at 200 mg/mL with *SI Appendix*, Fig. S10). This artifact was not seen in any of the other simulations and does not appear to strongly affect the diffusivity and viscosity of the UBQ solution (Figs. 2 and 3). The translational and rotational diffusion coefficients at high concentration (200 mg/mL) vary slightly for higher protein numbers in the simulation box. The absolute deviations in **7** and *SI Appendix*, Eq. **S10**). On the other hand, we observe a significant tail to high cluster numbers in the cluster distribution in these systems (*SI Appendix*, Fig. S10). Whereas the mean cluster sizes in the small and large systems are similar (Fig. 3*C*), occasional large clusters in the large systems appear to slightly suppress diffusion in a way not captured by the smaller simulation systems. Assessing and quantifying both effects in detail would merit a separate study.

Based on the clustering data for UBQ and GB3 solutions, we assume that the cluster size increases linearly with protein concentration up to intermediate (100 mg/mL) protein concentration. Although this simple model works surprisingly well in explaining our calculated diffusion data even at higher concentrations (Fig. 3 *A* and *B*), the actual functional dependence of cluster size on protein concentration may be nonlinear and depend on the specifics of the system (protein type, pH, ionic strength, temperature). Much longer simulations would be needed to precisely determine the cluster distribution and mean cluster size at high protein concentration.

The representation of dense protein solutions by Baxter’s sticky HS suspensions is limited to weak, short-range protein–protein interactions. As scattering data show (57), the sticky HS model does not represent experimental data well for strongly interacting particles or particles with significant attractive long-range (electrostatic) interactions. In our simulations and in cellular conditions, these interactions are effectively shielded by the ions in the solution, making this limitation less relevant in vivo. Nevertheless, it would be interesting to test the applicability of the cluster model for these cases.

## Conclusions

By performing all-atom molecular dynamics simulations of dense protein solutions, we found an increase in the viscosity of the solutions at higher protein volume fractions, consistent with experimental results (39, 61, 62). This increase is considerably higher than predicted by colloidal models of noninteracting HSs, stressing the importance of measuring or calculating rather than approximating the viscosity at protein volume fractions approaching cellular crowding conditions. We calculated translational and rotational diffusion coefficients and corrected them for finite-size effects using the respective viscosity of the solution. Translational diffusion and rotational diffusion are strongly affected by protein crowding. For LYZ solutions, experimentally measured diffusion coefficients are available also at high concentration and are in excellent agreement with our simulation data (Fig. 3 *A* and *B*). We calculated effective hydrodynamic radii using the Stokes–Einstein relations and found that a similar increase in the effective hydrodynamic radius can be inferred from the slowdown of translational and rotational diffusion caused by the formation of dynamic protein clusters. Indeed, establishing consistency with the Stokes–Einstein relations requires accounting for protein cluster formation (42) as a result of attractive interactions (60). Overall, we conclude that the concentration dependence of protein cluster size, the translational and rotational diffusion coefficient, and viscosity are consistent with each other (exception: LYZ at 100 mg/mL) and—for the proteins studied here—are explained well by Baxter’s sticky HS model of colloidal suspensions.

Representing the diffusion data in reduced form as a function of protein volume fraction showed that the relative slowdown in translational diffusion is consistent with results from previous studies. The relative slowdown in rotational diffusion shows a larger spread, consistent with the notion that rotational diffusivity depends more sensibly on clustering propensity and thus on the specifics of the protein interactions. Dynamic cluster formation has recently been observed also for membrane proteins (59) and shown to slow down rotational diffusion. In light of our analysis in terms of the cluster and colloidal models, we would expect similar affinities

We find that the proteins favor certain orientations for interactions and our findings on UBQ contact interfaces are consistent with experiments (82). The protein interactions lead to highly correlated motion at short distances and the correlation is sustained up to larger distances at low concentration. At high concentration, despite increased protein cluster formation, the pair correlation (19) at similar distances is decreased.

In the cellular environment, the situation is complicated by molecular heterogeneity, reactions, partitioning in microenvironments by phase separation, interactions with membranes and structural proteins, and other factors (99⇓⇓–102). Nevertheless, the findings here and in earlier work (103), as well as the observation that diffusion in cell lysates is similar to diffusion in crowded protein solutions (22), suggest that both in concentrated solution and in cells, proteins appear to diffuse not as isolated particles, but as members of dynamic clusters between which they constantly exchange. From the consistency of our diffusivity results with experiments in solution and on the basis of our cluster model, we conclude that—in crowded conditions corresponding to the cellular concentration—the strength of nonspecific protein–protein interactions for abundant proteins such as UBQ should correspond to low-millimolar binding.

We can now carry out atomistic MD simulations of crowded simulations at an unprecedented scale (29), here with up to 540 proteins and 3.6 million atoms in the box simulated over microseconds. Atomistic simulations of solutions of protein mixtures, possibly reflecting the distribution of proteins in the cell, no longer seem out of reach (52). Developments in nucleic acid force fields (104⇓⇓⇓⇓–109) will make it attractive to test the above findings on dense nucleic acid solutions and dense protein–nucleic acid mixtures. Ultimately, the macromolecular diversity in the cell will have to be considered (101) to predict passive diffusion in vivo.

### SI Appendix.

*SI Appendix* contains supplementary text; *SI Appendix*, Figs. S1–S21; *SI Appendix*, Tables S1–S3; Movie S1; and SI references.

### Movie S1.

Shown is an atomistic MD simulation of 540 GB3 proteins in concentrated solution (200 mg/mL) at simulation time 0–500 ns. The fully flexible proteins are shown in surface representation and differentiated by color. For clarity, water and ions are omitted. Proteins that seem to appear and disappear traverse the periodic boundaries.

## Materials and Methods

### MD Simulations of Dense Protein Solutions.

We performed all-atom MD simulations of solutions of human UBQ [PDB code 1UBQ (110)], GB3 [PDB code 1P7F (111)], LYZ [PDB code 1E8L (112)], and VIL [PDB code 1VII (113)] at up to five different densities with *SI Appendix*, Table S1). To mimic an infinitely dilute system, MD simulations with a single protein copy were carried out. The simulation procedures are detailed in *SI Appendix*.

### Viscosity Calculation and Approximations.

The low-frequency, low-shear viscosity *SI Appendix*. The dependence of η on the protein volume fraction ϕ is well captured by a quadratic function,

We compared the calculated viscosities to predictions from colloid theory. At low solute volume fractions ϕ, Einstein (66) predicted a linear dependence of the viscosity of HS suspensions on ϕ,

In addition, we also estimated the viscosity by assuming that the Stokes–Einstein relations for rotational diffusion,

### Translational Diffusion.

Translational diffusion coefficients *SI Appendix*. The diffusion coefficients were corrected for finite-size effects using (115)

We compared the calculated translational diffusion coefficients to the predictions of a dynamic cluster model without any free parameters,*SI Appendix* and calculated directly from the MD structures. This model is based on the assumption that the slowdown in translational diffusion is linked to the increase in the effective hydrodynamic radius and in the viscosity via the Stokes–Einstein relation for translational diffusion. We also compared the reduced translational diffusion coefficients *SI Appendix*).

### Rotational Diffusion.

Rotational diffusion coefficients were calculated following the procedure by Linke et al. (80, 116), as detailed in *SI Appendix*. Additionally, an effective rotational diffusion coefficient was obtained from fits to the orientational correlation function *SI Appendix*.

We compared the calculated rotational diffusion coefficients to the predictions of the dynamic cluster model,*SI Appendix*).

### Hydrodynamic Radius.

We solved the Stokes–Einstein relations for translational and rotational diffusion to define effective hydrodynamic radii for translation,

## Acknowledgments

We thank Jürgen Köfinger and Martin Vögele for helpful discussions and technical assistance and Kara Grotz for help with the simulation setup. This research was supported by the Max Planck Society (S.v.B., M.S., M.L., and G.H.), the Human Frontier Science Program RGP0026/2017 (S.v.B. and G.H.), and the Landes-Offensive zur Entwicklung Wissenschaftlich-ökonomischer Exzellenz (LOEWE) Dynamem program of the state of Hesse (M.S. and G.H.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: gerhard.hummer{at}biophys.mpg.de.

Author contributions: S.v.B. and G.H. designed research; S.v.B. and M.S. performed research; S.v.B., M.S., M.L., and G.H. analyzed data; and S.v.B., M.S., M.L., and G.H. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. H.-X.Z. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1817564116/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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_{2.5}in 2011, with a societal cost of $886 billion, highlighting the importance of modeling emissions at fine spatial scales to prioritize emissions mitigation efforts.

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