Fig. 3.

Protein diffusion and clustering. (A and B) Concentration-dependent translational (A) and rotational (B) protein diffusion. (A) Dependence of the translational diffusion coefficient on protein volume fraction $\varphi$. Solid circles and diamonds show the finite-size corrected diffusion coefficients $D_t$ of the small systems ($N\le 20$) and large systems ($N\ge 120$), respectively. Open squares show Hydropro calculations (70), and other open symbols show experimental data. UBQ data: $△$ (71), $▽$ (72). LYZ data: $△$ (73), $▽$ (74), $⊲$ (75), $⊳$ (21), $\star$ (76), $+$ (24), $\times$ (25). Lines show the predictions from the dynamic cluster model Eq. 8 with fitted $D_{t,\varphi =0}$. (B) Dependence of the rotational diffusion coefficients on protein volume fraction $\varphi$. Solid circles and diamonds show the diffusion coefficients obtained from the anisotropic diffusion tensor and corrected for finite-size effects (${\overline{D}}_r$) for the small systems ($N\le 20$) and large systems ($N\ge 120$), respectively. Stars ($\star$) and plus signs ($+$) show the finite-size corrected diffusion coefficients ${\tilde{D}}_r$ from integration of $⟨⟨P_1\left(\cos \theta \right)⟩⟩$ for the small and large systems, respectively. Triangles ($△$) represent NMR data for dilute UBQ (blue) (77), dilute GB3 (orange) (78), and dilute LYZ (green) (73) solution. Green crosses ($\times$) show NMR data for LYZ (25). Open squares show results from Hydropro (70) calculations. Lines show the predictions from the dynamic cluster model Eq. 9 with fitted $D_{r,\varphi =0}$. B, Inset shows a zoom-in at high protein volume fraction. (C and D) Protein-cluster model. (C) Dependence of mean cluster size $\overline{m}\left(\varphi \right)$ on protein volume fraction $\varphi$ [circles for small systems ($N\le 20$), diamonds for large systems ($N\ge 120$)]. The dashed line shows a linear fit $1+\zeta \varphi$ to the data at 0–100 mg/mL with binding propensity $\zeta$ listed in Table 1. Purple circles and diamonds are results from a Monte Carlo simulation of Baxter’s sticky HSs with $N=16$ and $N=120$ particles in the box, respectively, using $\tau =0.15$ and finite-range attractive interactions up to 1.05 $\sigma$. The dotted curve shows the fit $\overline{m}\left(\varphi \right)=1+\varphi /\tau +743.3{\varphi }^3$ to the MC data up to $\varphi =0.15$. Green crosses show experimental data on LYZ cluster formation (79). (C, Inset) Representative simulation snapshot of dense UBQ solution (100 mg/mL). Blue lines indicate the periodically replicated simulation boxes. Colors and colored labels indicate transient UBQ protein clusters and cluster size $m$, respectively (solvent not shown). (D) Dependence of reduced hydrodynamic radii cubed, ${R_h}^3/{R_{h,\varphi =\hspace{0.17em}0}}^3$, on protein volume fraction $\varphi$. The effective $R_h$ is calculated from the Stokes–Einstein relations for translation, Eq. 10 (circles), and for rotation, Eq. 11 (squares). The dashed lines show the prediction $1+\zeta \varphi$ from C with $\zeta$ values from Table 1.