Universal Relation between Instantaneous Diffusivity and Radius of Gyration of Proteins in Aqueous Solution

Protein conformational fluctuations are highly complex and exhibit long-term correlations. Here, molecular dynamics simulations of small proteins demonstrate that these conformational fluctuations directly affect the protein’s instantaneous diffusivity $D_I$. We find that the radius of gyration $R_g$ of the proteins exhibits $1/f$ fluctuations that are synchronous with the fluctuations of $D_I$. Our analysis demonstrates the validity of the local Stokes-Einstein–type relation $D_I\propto 1/(R_g+R_0)$, where $R_0\sim 0.3\mathrm{nm}$ is assumed to be a hydration layer around the protein. From the analysis of different protein types with both strong and weak conformational fluctuations, the validity of the Stokes-Einstein–type relation appears to be a general property.

Figure 1

Conformational fluctuations of Chignolin at 310 K and 0.1 MPa. (a) Time series of the gyration radius $R_g$. Thin and thick lines represent the unsmoothed original values every 1 ns and a smoothed moving average with 100 ns averaging window, respectively. (b) Probability density function of $R_g$. (c) Ensemble-averaged PSD of $R_g$ averaged over five trajectories of $40\mu \mathrm{s}$. Solid lines are shown for reference. (d) Ensemble- and time-averaged mean squared protein end-to-end distance for measurement time $t=40\mu \mathrm{s}$. (e) Ensemble-averaged PSDs of the end-to-end distance. Different colored symbols represent the PSDs for different measurement times.

Figure 2

Decomposition of the relaxation modes of Chignolin at 310 K and 0.1 MPa. (a) Free energy map for the slowest relaxation mode (mode 1) vs the second slowest mode (mode 2) obtained by RMA using the coordinates of $\mathrm{C}\alpha$ atoms with parameters $t_0=0.5$ and $\tau =0.1\mathrm{ns}$. Snapshots of protein conformations corresponding to the free energy maps: (i) native state, (ii) metastable state, and (iii)–(vii) states 3–7. Residues 1–10 are colored green to blue. (b) Ensemble-averaged cumulative PSDs of relaxation modes and principal components. RMA and principal component analysis (PCA) were performed using coordinates of heavy atoms or $\mathrm{C}\alpha$ atoms. Parameters for RMA were set as RMA0 ($t_0=0$ and $\tau =0.1\mathrm{ns}$) and RMA5 ($t_0=0.5$ and $\tau =0.1\mathrm{ns}$).

Figure 3

Fluctuating diffusivity of Chignolin at 310 K and 0.1 MPa. (a) Normalized magnitude ${\widehat{\mathrm{\Phi }}}_1(\mathrm{\Delta },t)$ and orientation ${\widehat{\mathrm{\Phi }}}_2(\mathrm{\Delta },t)$ correlation functions. Forty-five divided trajectories were used with a lag time $\mathrm{\Delta }=50\mathrm{ps}$. (b) Correlation between the mean $R_g$ and the instantaneous diffusion coefficient $D_I$ in each diffusive state. (c) Time series of $R_g$ and temporal diffusion coefficient (TDC). Thin lines represent unsmoothed original values every 10 ns. Thick lines represent mean $R_g$ and $D_I$ in each state, where $t=100\mathrm{ns}$ and $\mathrm{\Delta }=10\mathrm{ps}$ were used to obtain the TDC.

Figure 4

Fluctuating diffusivity of Chignolin at different temperature and pressure conditions: (a) 280 K, 0.1 MPa and (b) 400 K, 400 MPa. Left: time series of $R_g$. Thin and thick lines represent unsmoothed original values every 1 ns and smoothed moving average with 100 ns averaging window, respectively. Middle: normalized magnitude ${\widehat{\mathrm{\Phi }}}_1(\mathrm{\Delta },t)$ and orientation ${\widehat{\mathrm{\Phi }}}_2(\mathrm{\Delta },t)$ correlation functions. Thirty-five divided trajectories were used with a lag time $\mathrm{\Delta }=50\mathrm{ps}$. Right: correlation between mean $R_g$ and $D_I$ in each diffusive state, where $t=100\mathrm{ns}$ and $\mathrm{\Delta }=10\mathrm{ps}$ were used to obtain the TDC.