Simple model of $p\text{H}$-induced protein denaturation

The $p\text{H}$-induced conformational changes of proteins are systematically studied in the framework of a hydrophobic-polar (HP) model, in which proteins are dramatically simplified as chains of hydrophobic (H) and polar (P) beads on a lattice. We express the electrostatic interaction, the principal driving force of $p\text{H}$-induced unfolding that is not included in the conventional HP model, as the repulsive energy term between P monomers. As a result of the exact enumeration of all of the 14- to 18-mers, it is found that lowest-energy states in many sequences change from single “native” conformations to multiple sets of “denatured” conformations with an increase in the electrostatic repulsion. The switching of the lowest-energy states occurs in quite a similar way to real proteins: it is almost always between two states, while in a small fraction of $\ge 16$-mers it is between three states. We also calculate the structural fluctuations for all of the denatured states and find that the denatured states contain a broad range of incompletely unfolded conformations, similar to “molten globule” states referred to in acid or alkaline denatured real proteins. These results show that the proposed model provides a simple physical picture of $p\text{H}$-induced protein denaturation.

Figure 1

Folding behavior of the two-state folder HHPHHPPPPPHPHPPHPH. (a) Thermal average of the structural compactness $\langle \rho \rangle$ as the function of $\mu$, the relative strength of the electrostatic repulsion, for several different temperatures. (b) Energy diagram for all possible conformations. The lowest-energy state changes from the N (native) state [thick solid (blue) line] to D (denatured) state [dashed (red) line] at $\mu =1/2$. (c) Fractional population of the N state $\left(P_{\mathrm{N}}\right)$ and the D state $\left(P_{\mathrm{D}}\right)$. $P_{\mathrm{N}}=P(7,2)$ and $P_{\mathrm{D}}=P(6,0)$ [see Eq. (6)]. The temperature is set to $|{\epsilon }_{\mathrm{HH}}|/kT=10$. (d) N conformation. (e) D conformations. The filled circles represent the H (hydrophobic) residues and the open circles represent the P (polar) residues. The values of the compactness are also indicated. The N state contains only a single N conformation while the D state consists of three D conformations.

Figure 2

Folding behavior of the three-state folder PHHPPPPPHHPPPPPHHP. (a) Thermal average of the structural compactness $\langle \rho \rangle$ as the function of $\mu$ for several different temperatures. The three-step transition can be seen. (b) Energy diagram for all possible conformations. The lowest-energy state changes from the N state [thick solid (blue) line] to ${\mathrm{D}}_1$ state [dotted and dashed (green) line] at $\mu =1/4$, and from the ${\mathrm{D}}_1$ state to ${\mathrm{D}}_2$ state [dashed (red) line] at $\mu =1/2$. (c) Fractional population of the N state $\left(P_{\mathrm{N}}\right)$ and two D states $(P_{{\mathrm{D}}_1}$ and $P_{{\mathrm{D}}_2})$. $P_{\mathrm{N}}=P(4,6),P_{{\mathrm{D}}_1}=P(3,2)$ and $P_{{\mathrm{D}}_2}=P(2,0)$ [see Eq. (6)]. The temperature is set to $|{\epsilon }_{\mathrm{HH}}|/kT=10$. (d) N conformation. (e) Examples of ${\mathrm{D}}_1$ conformations. (f) Examples of ${\mathrm{D}}_2$ conformations. The filled circles represent the H (hydrophobic) residues and the open circles represent the P (polar) residues. The N state contains only a single N conformation, while the ${\mathrm{D}}_1$ and ${\mathrm{D}}_2$ states consist of 69 and 144 D conformations, respectively.

Figure 3

Thermal average of the structural compactness $\langle \rho \rangle$ for (a)–(c) the two-state folders, (d)–(f) representative sequences as a function of $\mu$. The temperature is set to $|{\epsilon }_{\mathrm{HH}}|/kT=10$. The values of ${\mu }^{*}$ are also shown for the two-state folders (see text).

Figure 4

Properties of the D states of the two-state folders. (a) Relationship between the compactness of the N states $\left({\rho }_{\mathrm{N}}\right)$ and D states $\left({\rho }_{\mathrm{D}}\right)$. They are related to $\langle \rho \rangle$ by ${\rho }_{\mathrm{N}}={lim}_{\mu \to 0}\langle \rho \rangle$ and ${\rho }_{\mathrm{D}}={lim}_{\mu \to \infty }\langle \rho \rangle$ at $T=0$. The dotted line indicates ${\rho }_{\mathrm{D}}={\rho }_{\mathrm{N}}$. (b) Structural fluctuation of the D states $\left(\delta {\rho }_{\mathrm{D}}\right)$.

Figure 5

Evolution of the structural fluctuation $\delta \rho$ for (a) HHPHHPPPPPHPHPPHPH and (b) PHHPPPPPHHPPPPPHHP. The superpositions of $\delta \rho$ on $\langle \rho \rangle$ are also shown (bottom). The temperature is set to $|{\epsilon }_{\mathrm{HH}}|/kT=10$.