Universality and Specificity in Protein Fluctuation Dynamics

We investigate the universal scaling of protein fluctuation dynamics with a site-specific diffusive model of protein motion, which predicts an initial subdiffusive regime in the configurational relaxation. The long-time dynamics of proteins is controlled by an activated regime. We argue that the hierarchical free energy barriers set the time scales of biological processes and establish an upper limit to the size of single protein domains. We find it compelling that the scaling behavior for the protein dynamics is in close agreement with the Kardar-Parisi-Zhang scaling exponents.

Figure 1

Mode-dependent free energy barrier, $E_p^{†}={(\epsilon L_p)}^{\gamma }$, mode length scale $L_p$ for all proteins with $\gamma =0.93\pm 0.20$ and $\epsilon =6.5(\mathrm{kcal}{\mathrm{mol}}^{-1}{)}^{1/\gamma }{\mathrm{nm}}^{-1}$ (blue dashed line).

Figure 2

Left: Mode length $L_p$ and mode index for all proteins and the scaling $L_p=L_{\max }{p}^{-\beta }$ with $\beta =0.41$ (blue dashed line). Right: Predicted subdiffusive mean-squared displacement $⟨R^2(t)⟩\propto t^{\nu }$ with $\nu =0.26$ (blue dashed line), the average MSD of all proteins (solid line), and the MSD of each protein individually.

Figure 3

Reduced mode time scales, $\tau /(\eta \beta l^3)$, for all proteins as a function of mode length scale, $L_p$, where $\eta$, $\beta$, and $l$ are solvent viscosity, inverse temperature, and average effective bond length, respectively. The red triangles show the reduced mode time scales before free energy barrier correction and the scaling ${\tau }_{0,p}=C_{\tau }{L}_p^{\alpha }$ with $\alpha =2.00\pm 0.41$ (blue line). Free-energy barrier corrected, reduced mode time scales (black circles) and the scaling ${\tau }_p=C_{\tau }{L}_p^{\alpha }\exp [(\epsilon L_p{)}^{\gamma }/k_{B}T]$ (magenta line). Reduced time scales of folding for 52 proteins, plotted against the largest mode length scale, approximately the protein radius of gyration (green circles) [6].

Figure 4

Mean squared displacement of the ubiquitin protein at 300 (left panel) and 390 K (right panel) from the ms scale simulations of Piana et al. (solid line) [24]. The short-time subdiffusive $\mathrm{MSD}\propto t^{\nu }$ with $\nu =0.26$ (dashed line), and the long-time crossover to the activated regime from Eq. (5) (dot-dashed line).