Protein folding as a jamming transition

Abstract

Proteins fold to a specific functional conformation with a densely packed hydrophobic core that controls their stability. We develop a geometric, yet all-atom model for proteins that explains the universal core packing fraction of ${\varphi }_c=0.55$ found in experimental measurements. We show that as the hydrophobic interactions increase relative to the temperature, a novel jamming transition occurs when the core packing fraction exceeds ${\varphi }_c$. The model also recapitulates the global structure of proteins since it can accurately refold to native-like structures from partially unfolded states.

FIG. 1.

The nonbonded dimensionless pair potential ${\tilde{V}}_{ij}=V_{ij}/{ϵ}_r$ plotted versus atomic separation ${\tilde{r}}_{ij}=r_{ij}/{\sigma }_{ij}$ for purely-repulsive interactions (Eq. 1) (solid line) and attractive interactions (Eq. 5) (dashed line) and (b) the corresponding dimensionless force ${\tilde{F}}_{ij}=F_{ij}{\sigma }_{ij}/{ϵ}_r$. The symbols represent the onset of repulsive interactions where ${\tilde{r}}_{ij}=1$ and ${\tilde{V}}_{ij}=-{\tilde{V}}_c=-V_c/{ϵ}_r$ (circles), the change in spring constant where ${\tilde{r}}_{ij}={\tilde{r}}_{\beta }=1+r_{\beta }{\sigma }_{ij}$ and ${\tilde{F}}_{ij}=-{\tilde{\beta }}_{ij}=-\beta {\lambda }_{ij}{\sigma }_{ij}/{ϵ}_r$ (squares), and the separation above which the interactions are zero ${\tilde{r}}_{ij}={\tilde{r}}_{\alpha }=1+r_{\alpha }/{\sigma }_{ij}=1+\alpha$ (triangles). (c) The difference $\mathrm{\Delta }f$ between the average fraction of backbone dihedral angle outliers (black circles) and side chain dihedral angle outliers (grey squares) between the HS model and proteins from a high-quality x-ray crystal structure database plotted versus the temperature $T/{ϵ}_r$ at which the HS model proteins were simulated. Inset: The probability distribution of backbone dihedral angles $P(\phi ,\psi )$ sampled by high-quality x-ray crystal structures of proteins. The colors from light to dark indicate increasing probability on a logarithmic scale. (d) Probability distribution of the average core packing fraction $P(⟨\varphi ⟩)$ in high-quality x-ray crystal structures of proteins calculated using the optimized HS atom sizes on a semi-log plot with a Gaussian fit (black dashed line).

FIG. 2.

(a) The average core packing fraction $⟨\varphi ⟩$ plotted versus the attraction strength ${\alpha }^2\beta$ for the HS+HP protein model for temperatures $T/{ϵ}_r={10}^{-6}$ (yellow), 10⁻⁷ (green), and 10⁻⁸ (blue) and $\alpha =0.5$ (circles), 1.0 (squares), 1.5 (upward triangles), and 2.0 (downward triangles). The horizontal red dot-dashed line and cyan shading indicate the average and standard deviation of the core packing fraction in the high-resolution x-ray crystal structure data set. The black dashed lines indicate fits to Eq. 6. (b) The average repulsive potential energy per atom $⟨V_r/N⟩$ plotted versus ${\alpha }^2\beta$. The black dashed lines indicate fits to Eq. 7. (c) $⟨V_r/N⟩$ plotted versus $⟨\varphi ⟩$. The vertical red dot-dashed line and cyan shading indicate the average and standard deviation of the core packing fraction in the high-resolution x-ray crystal structure data set. The black dashed lines indicate fits to Eq. 8.

FIG. 3.

(a) The vibrational density of states (VDOS) $D\left({\omega }_n\right)$, where ${\omega }_n$ is the frequency, of the ${\mathrm{C}}_{\alpha }$ atoms in the HS+HP model at $T/{ϵ}_r={10}^{-8}$ for all $\alpha$ and $\beta$ in Fig. 2. (b) Participation ratio $p_r\left({\omega }_n\right)$ plotted versus ${\omega }_n$. The average total nonbonded repulsive potential energy per atom $⟨V_r/N⟩$ increases from blue to red on a logarithmic scale.

FIG. 4.

(a) ${\mathrm{C}}_{\alpha }\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{D}⟨{\mathrm{\Delta }}_f⟩$ in Å between the HS+HP model proteins and the x-ray crystal structures averaged over 20 proteins plotted versus ${\alpha }^2\beta$ when starting from the experimental structure for temperature $T/{ϵ}_r={10}^{-6}$ (yellow), 10⁻⁷ (green), and 10⁻⁸ (blue) and $\alpha =0.5$ (circles), 1.0 (squares), 1.5 (upward triangles), and 2.0 (downward triangles). (b) Average ${\mathrm{C}}_{\alpha }\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{D}⟨{\mathrm{\Delta }}_f⟩$ plotted versus the initial ${\mathrm{C}}_{\alpha }\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{D}{\mathrm{\Delta }}_i$ in $\beta$ for $T/{ϵ}_r={10}^{-7}$. The filled circles are colored by $\alpha =0.5-5.5$ increasing from purple to yellow, and $\beta$ is set so that ${\alpha }^2\beta ~T/{ϵ}_r$. All-atom MD simulations of a single protein (PDBID: 2IGP) using the Amber99SB-ILDN force field are shown as grey squares. The red dashed line indicates $⟨{\mathrm{\Delta }}_f⟩={\mathrm{\Delta }}_i$.