Critical Fluctuations in the Native State of Proteins

Based on protein structural ensembles determined by nuclear magnetic resonance, we study the position fluctuations of residues by calculating distance-dependent correlations and conducting finite-size scaling analysis. The fluctuations exhibit high susceptibility and long-range correlations up to the protein sizes. The scaling relations between the correlations or susceptibility and protein sizes resemble those in other physical and biological systems near their critical points. These results indicate that, at the native states, motions of each residue are felt by every other one in the protein. We also find that proteins with larger susceptibility are more frequently observed in nature. Overall, our results suggest that the protein’s native state is critical.

Figure 1

(a) Example of a structural ensemble (protein PDB code: 1BAK) showing the 20 structures (grey) aligned to the reference structure (orange). (b) The average structure (red) and the $q$th structure (cyan) for the protein in (a). The displacement $\mathrm{\Delta }{\overrightarrow{r}}_{i,q}$ (or $\mathrm{\Delta }{\overrightarrow{r}}_{j,q}$) of the $i$th (or $j$th) residue from its counterpart in the average structure is marked with black arrows. The distance $r_{ij}$ between residues $i$ and $j$ is marked with red dashes. (c) The magnitude of the residues’ position fluctuations $|\mathrm{\Delta }{\overrightarrow{r}}_{i,q}|$ for the $q$th structure ($q=1,2,\dots$) in the ensemble.

Figure 2

Correlation $\varphi (r)$: (a) $\varphi (r)$ for proteins with various structural features: 2I94 ($\alpha$ class), 1KIK ($\beta$ class), 1W4U ($\alpha +\beta$), 1TI3 ($\alpha /\beta$), 1QXF (small), 1HCW (designed), 2M3U (multidomain), and 1HV2 (multichain). The correlation length $\xi$ (for 1HCW) is indicated by an arrow. (b) $\varphi (r)$ for proteins with similar $R_g\approx 12\AA$. (c) Scattering plot of $\xi$ and $R_g$ for all 4988 proteins. The red open circles show the average $⟨\xi ⟩$ for proteins with similar $R_g$. (d) $\varphi (r)$ for the subsets of proteins in bins labeled by $R_g=7,9,\dots ,23\AA$ (see text). (e) Scaling plot of $\varphi (\widehat{r})$ with $\widehat{r}=r/\xi$ for curves in (d). (f) $\varphi (r)$ and covariance $C(r)$ (inset) related to specific types of residue pairs for proteins with $R_g\approx 15\AA$.

Figure 3

Finite-size scaling. (a) Susceptibility $\chi$ vs shape factor $s$ for proteins with different sizes $N$. The peak height ${\chi }_m$ and position $s_m$ show a power-law relation ${\chi }_m\sim (s_m-s_c{)}^{-\gamma }$ (thick red line). (b) Scaling of peak heights with $N$ as ${\chi }_m\sim N^{-\alpha \gamma /\mu }$. (c) Correlation length $\xi$ vs protein size $N$ and $R_g$ vs $N$ (inset). (d) Shape factor $s$ vs $N$. (e) Susceptibility $\chi$ vs shape factor $s$. (f) Susceptibility $\chi$ vs protein size $N$. Error bars are standard errors of the mean.

Figure 4

(a) Shape factor distribution for all proteins. (b) The most frequent shape factor $s_{\max }$ vs the position of the maximal susceptibility $s_m$ for subsets of proteins with similar $R_g$.