Theoretical analysis on thermodynamic stability of chignolin

Abstract

Understanding the dominant factor in thermodynamic stability of proteins remains an open challenge. Kauzmann's hydrophobic interaction hypothesis, which considers hydrophobic interactions between nonpolar groups as the dominant factor, has been widely accepted for about sixty years and attracted many scientists. The hypothesis, however, has not been verified or disproved because it is difficult, both theoretically and experimentally, to quantify the solvent effects on the free energy change in protein folding. Here, we developed a computational method for extracting the dominant factor behind thermodynamic stability of proteins and applied it to a small, designed protein, chignolin. The resulting free energy profile quantitatively agreed with the molecular dynamics simulations. Decomposition of the free energy profile indicated that intramolecular interactions predominantly stabilized collapsed conformations, whereas solvent-induced interactions, including hydrophobic ones, destabilized them. These results obtained for chignolin were consistent with the site-directed mutagenesis and calorimetry experiments for globular proteins with hydrophobic interior cores.

Figure 1

Calculation scheme of free energy profile of a protein. The distance between alpha carbon atoms at the C-terminus and N-terminus (shown as yellow spheres) is introduced as the coordinate R specifying the dimensions of the protein. (a) Thermodynamic cycle depicting the relationship in Eq. 1 among the free energy profile of the protein in water, F(R), the free energy profile in vacuum, F_vac(R), and the excess chemical potential profile, μ_ex(R), at a distance R. (b) Two different thermodynamic cycles, which allow for the calculation of F(R) and F_vac(R), if the following are obtained: free energy profile determined by the generalized Born (GB) model umbrella sampling MD simulations with a dielectric constant ε_r = 80, F_GB(R), excess chemical potential profile of the protein in the GB model with ε_r = 80 (Eq. 4), ${\mu }_{ex}^{\mathrm{GB}}(R)$, and free energy difference between the protein in water described by the GB model and that described by the RMDFT model (Eq. 7), $\mathrm{\Delta }{\mu }_{\mathrm{DFT}}^{\mathrm{GB}}(R)$.

Figure 2

Free energy profiles of chignolin in water at 298 K. All profiles are shown in k_BT and are shifted vertically, so that the value becomes zero at a distance of R = 0.5 nm, which corresponds to the native state. (a) Free energy profile calculated according to the GB model, F_GB(R), and that corrected by the RMDFT method, F(R). (b) Free energy profile in vacuum, F_vac(R), excess chemical potential profile, μ_ex(R), and nonpolar part of μ_ex(R), μ_nonpol(R), calculated by removing all partial charges on chignolin (see Computational Details). (c,d) These free energy differences from the native state (R = 0.5 nm) are shown for the misfolded state at R = 0.6 nm, the transition state (TS) at R = 1.0 nm, and an unfolded state (UFS) at R = 1.8 nm. The numbers beside the legends in (b) indicate the vertically shifted value for these profiles. Here and hereafter, the error bars for all profiles indicate the standard error. The shown ternary structures are for the native state from the Protein Data Bank database (PDB: 1UAO), misfolded state obtained at R = 0.6 nm, and unfolded state obtained at R = 1.8 nm. The yellow and red spheres depict the alpha carbon atoms at the C-terminus and N-terminus, respectively. In the misfolded state, the relative position of Try-9 compared with Try-2 is different from that of the native state because of rotation in backbone torsion angle ψ for Gly-7.

Figure 3

Comparison of free energy profiles for chignolin in water at 298 K and 373 K. (a) Free energy profile calculated according to the GB model, F_GB(R), and that corrected by the RMDFT method, F(R). (b) Difference between the free energy profiles at 373 K and 298 K, ${\mathrm{\Delta }}_{T}F(R)=F(R,373\mathrm{K})/k_{B}T-F(R,298\mathrm{K})/k_{B}T$ and the corresponding differences for the two components, ${\mathrm{\Delta }}_T{F}_{vac}(R)$ and ${\mathrm{\Delta }}_T{\mu }_{ex}(R)$, where ${\mathrm{\Delta }}_{T}F(R)={\mathrm{\Delta }}_T{F}_{vac}(R)+$${\mathrm{\Delta }}_T{\mu }_{ex}(R)$. (c) Nonpolar part and polar part for ${\mathrm{\Delta }}_T{\mu }_{ex}(R)$, ${\mathrm{\Delta }}_T{\mu }_{nonpol}(R)$ and ${\mathrm{\Delta }}_T{\mu }_{pol}(R)$, respectively, where ${\mathrm{\Delta }}_T{\mu }_{ex}(R)={\mathrm{\Delta }}_T{\mu }_{nonpol}(R)+{\mathrm{\Delta }}_T{\mu }_{pol}(R)$. (d) Free energy profile in vacuum, F_vac(R), and its energy part, $E_{vac}^{intra}(R)$, at 298 K and 373 K. The difference between F_vac(R) and $E_{vac}^{intra}(R)$ corresponds to the entropic term of F_vac(R), $-T{S}_{vac}^{intra}(R)=F_{vac}(R)-E_{vac}^{intra}(R)$ (see Eq. 2).

Figure 4

Effect of pressure on the free energy profile F(R). (a) Pressure dependence of the free energy profile at 298 K. (b) Difference in the free energy between 8000 bar and 1 bar, ${\mathrm{\Delta }}_{P}F(R)=F(R,8000\mathrm{bar})/k_{B}T-F(R,1\mathrm{bar})/k_{B}T$. Shown are the excess chemical potential difference, ${\mathrm{\Delta }}_P{\mu }_{ex}(R)$, the nonpolar part ${\mathrm{\Delta }}_P{\mu }_{nonpol}(R)$, and the electrostatic part, ${\mathrm{\Delta }}_P{\mu }_{pol}(R)={\mathrm{\Delta }}_P{\mu }_{ex}(R)-{\mathrm{\Delta }}_P{\mu }_{nonpol}(R)$, resulting from the independence of F_vac(R) on pressure.