Critical Phenomena in the Temperature-Pressure-Crowding Phase Diagram of a Protein

Inside the cell, proteins fold and perform complex functions through global structural rearrangements. For proper function, they need to be at the brink of instability to be susceptible to small environmental fluctuations yet stable enough to maintain structural integrity. These apparently conflicting properties are exhibited by systems near a critical point, where distinct phases merge. This concept goes beyond previous studies that propose proteins have a well-defined folded and unfolded phase boundary in the pressure-temperature plane. Here, by modeling the protein phosphoglycerate kinase (PGK) on the temperature ($T$), pressure ($P$), and crowding volume-fraction ($\varphi$) phase diagram, we demonstrate a critical transition where phases merge, and PGK exhibits large structural fluctuations. Above the critical temperature ($T_c$), the difference between the intermediate and unfolded phases disappears. When $\varphi$ increases, the $T_c$ moves to a lower $T$. With experiments mapping the $T\text{−}P\text{−}\varphi$ space, we verify the calculations and reveal a critical point at 305 K and 170 MPa that moves to a lower $T$ as $\varphi$ increases. Crowding shifts PGK closer to a critical line in its natural parameter space, where large conformational changes can occur without costly free-energy barriers. Specific structures are proposed for each phase based on the simulation.

Popular Summary

Many proteins fold into a well-packed structure in order to function, yet the folded phase must remain sufficiently flexible. It is unclear how proteins satisfy these contradictory constraints, especially in the crowded environment of a cell. We propose that these properties can coexist by tuning where the protein operates within its temperature-pressure-crowding phase diagram. Phase diagrams may contain “critical points” where the difference between any two phases disappears, such as when liquid and vapor water become indistinguishable. We show that a protein can also have such a critical point. The enzyme phosphoglycerate kinase, which is involved in producing the energy molecule of the cell, turns out to have a very crowding-sensitive critical point, above which the protein forms new structures.

To paint a complete picture of the critical point for this protein, we expand upon the conventional temperature-pressure folding phase diagram by varying the amount of crowding from surrounding macromolecules using computational simulations, a simple statistical mechanical model, and experiments. From simulations, we observe an intricate phase diagram, which contains a critical point that moves to a lower temperature as the crowding increases. We complement our simulation results by deriving an equation for the critical line in the entire phase diagram. To test our computational model, we observe folding transitions of phosphoglycerate kinase by fluorescence experiments, which validate the predicted critical point behavior.

Our findings suggest that being near a critical point can be advantageous for a protein’s biological function.

Figure 1

PGK surrounded by crowders and the desolvation potential between residues. (a) A snapshot from the coarse-grained molecular simulation of PGK’s spherical compact state (SPH) surrounded by crowders (gray) at the volume fraction of 40%. The N and C domain and hinge are in red, blue, and yellow, respectively. (b) The pressure-dependent desolvation potential at ${\sigma }^{3}P/{\epsilon }_0=4.6$, 6.6, and 9.2 (${\epsilon }_0/{\sigma }^3\approx 76\mathrm{MPa}$) contains a desolvation barrier with a width ($|r^{\prime }-r^{\prime \prime }|$) the size of a water molecule (blue). This barrier incorporates the entropic cost of expelling a solvent molecule between two residues (gold). The Lennard-Jones potential is plotted in light gray for comparison.

Figure 2

Solvation and crowding give rise to an intricate phase diagram of PGK. (a),(c) Schematics of PGK’s behavior in the crowding volume-fraction-pressure ($\varphi \text{−}P$) phase plane and (b),(d) corresponding free energy with respect to the overlap $\chi$ and crowding volume fraction $\varphi$ at the folding pressure at low (a),(b) and high (c),(d) temperatures. Solid lines represent the division between distinct configurational phases that are separated by a free-energy barrier from simulations at $\varphi =0$, 0.2, and 0.4 and pressures from ${\sigma }^{3}P/{\epsilon }_0={10}^3$ to 13 (${\epsilon }_0/{\sigma }^3\approx 76\mathrm{MPa}$). The dashed line (a) represents a continuous transition along $\varphi$, and red dots (a),(b) represent an approximate position of the critical points. The orange arrow (b) marks the peak of the barrier that diminishes until it disappears after the critical point. Collapsed crystal, spherical, and swollen unfolded states are indistinguishable in terms of free energy. These configurations are reconstructed from coarse-grained models to all-atomistic protein models for illustration purposes using SCAAL (side chain-$C_{\alpha }$ to all-atom method) [29]. The N and C domains and hinge are in red, blue, and yellow, respectively. A cyan sphere is inserted in between residues to show water-mediated contacts.

Figure 3

Experimental $T\text{−}P\text{−}\varphi$ phase diagram of PGK (full data in Ref. [10]). (a) The derivative of the mean tryptophan fluorescence wavelength vs the pressure of PGK at 282 K calculated from fluorescence spectra. Two of six Ficoll 70 concentrations are shown. The markers show the data points, and the solid line shows a cubic spline interpolation. The blue curve ($0\mathrm{mg}/\mathrm{ml}$ Ficoll 70) has two peaks as the pressure increases, signifying two transitions; the magenta curve ($200\mathrm{mg}/\mathrm{ml}$ Ficoll 70) has only one peak point, signifying only one transition when pressure is applied. The dashed lines point from transition midpoints to the corresponding point in the phase diagram. (b) $P\text{−}T$ phase diagrams at several $\varphi$ obtained by fitting the fluorescence data to obtain the inflection points of ${\lambda }_m(P)$ (peaks in the derivative $\partial {\lambda }_m/\partial P$). Three of six Ficoll 70 concentrations are shown. Circles represent midpoint pressures measured at 282, 288, 296, 303, 309, and 317 K in absence of Ficoll 70 ($0\mathrm{mg}/\mathrm{ml}$), asterisks represent transitions for the middle Ficoll 70 concentration ($100\mathrm{mg}/\mathrm{ml}$), and triangles represent transitions for the highest Ficoll 70 concentration ($200\mathrm{mg}/\mathrm{ml}$). At high $T$, or upon increasing the Ficoll 70 concentration, the second (higher $P$) transition disappears, mapping out a critical point that moves to a lower $T_c$ at a higher Ficoll 70 concentration. Solid elliptical curves going through the circles are fits to Eq. (4) representing the $\mathrm{\Delta }G=0$ curves. (c) Equivalent data as in (a) at 317 K. Note that the second (higher $P$) transition is never present at high $T$.

Figure 4

$T\text{−}P\text{−}\varphi$ phase diagram of PGK from the theory mapped onto the experimental data. (a) PGK’s $\varphi \text{−}P$ phase plane at high (magenta) and low (black) $T$. Dotted lines represent the division between distinct configurational phases. The red dot signifies a critical point. (b) Slices of the $P\text{−}T$ phase diagram observed experimentally at no Ficoll 70 (black curve) and $100\mathrm{mg}/\mathrm{ml}$ Ficoll 70 (cyan curve). Note that I-U coexistence curve terminates at the critical point (in red dots) and shifts $T_c$ to a lower temperature in the presence of Ficoll 70. Solid elliptical curves going through the circles are the fits representing the $\mathrm{\Delta }G=0$ curves. (c) A $T\text{−}P\text{−}\varphi$ phase diagram of PGK. The blue surface is the coexistence surface bordering the C phase (C-I, C-U, or C-SPH/CC/SPH depending on $T$ and $\varphi$), and the red surface is the I-U coexistence surface. The dashed magenta and black lines are the $\varphi \text{−}P$ cross section from Fig. 4(a), and the solid black and cyan lines are the $P\text{−}T$ cross section from Fig. 4(b). The bold, red line bordering the red surface is the critical line.

Figure 5

Cavity volume and structural fluctuations near the critical regime. (a) Cavity volume fluctuations $\delta V^2=⟨V^2⟩-⟨V{⟩}^2$ (or proportionally compressibility) of PGK at $k_{B}T/{\epsilon }_0=0.97$ and ${\sigma }^{3}P/{\epsilon }_0=4.6$, 6.6, and 9.2 with (orange line) and without (blue line) crowding (${\epsilon }_0/{\sigma }^3\approx 76\mathrm{MPa}$). (b) Overlap fluctuations $\delta {\chi }^2=⟨{\chi }^2⟩-⟨\chi {⟩}^2$ as a function of the covolume at pressure ${\sigma }^{3}P/{\epsilon }_0=6.6$. (c) Conformations from the ensemble in the presence of crowding at $k_{B}T/{\epsilon }_0=0.97$ and ${\sigma }^{3}P/{\epsilon }_0=6.6$ with covolumes approximately equal to $1.1\times {10}^5{\AA }^3$ and cavities approximately equal to $200{\AA }^3$. The leftmost conformation is a crystal state. The following structures from left to right have a $\chi =0.31$, 0.38, and 0.48. The N and C domains and hinge are in red, blue, and yellow, respectively. Covolumes are shown as translucent surfaces surrounding the protein, and cavity surfaces are shown in green.