Solvation of Proteins: Linking Thermodynamics to Geometry

We calculate the solvation free energy of proteins in the tube model of Banavar and Maritan [Rev. Mod. Phys. 75, 23 (2003)] using morphological thermodynamics which is based on Hadwiger’s theorem of integral geometry. Thereby we extend recent results by Snir and Kamien [Science 307, 1067 (2005)] to hard-sphere solvents at finite packing fractions and obtain new conclusions. Depending on the solvent properties, parameter regions are identified where the $\beta$ sheet, the optimal helix, or neither is favored.

Figure 1

Left: Helical conformation of a protein in the tube model. Right: A possible parallel surface for the same conformation. Because of self-intersection one or several helical intersection lines appear (in red; thicker lines) giving rise to additional contributions to $C$ and $X$, and hence to $F_{\mathrm{sol}}$.

Figure 2

Geometric measures $V$, $A$, and $C$ for a range of close-packed helical tube conformations, which have minimal $P_h$. Calculations are performed for the parallel body at different distances $R$ (radius of solvent hard core). The integral Gaussian curvature $X$ vanishes, consistent with the given topology. $R_h>(<)R_h^{*}$ corresponds to the TTCP (CCP) regime.

Figure 3

Diagram of the different protein structures which minimize $F_{\mathrm{sol}}$ in a hard-sphere solvent with packing fraction $\eta$ and solvent radius $R$. Transitions between the regions are either continuous (dashed lines) or discontinuous (full line) in $R_h$. In the limit $\eta \to 0$ there is no transition at $R=R_s$. The systems indicated by dots are discussed in Fig. 4.

Figure 4

$F_{\mathrm{sol}}(R_h)$ relative to its value at $R_h=\infty$ for the hard-sphere solvents indicated in Fig. 3, except P6. $F_{\mathrm{sol}}$ is shown for configurations along the line of close-packed helices with a focus on the turn-to-turn distance limited regime ($R_h>R_h^{*}$). In the curvature limited regime ($R_h<R_h^{*}$) the values $\Delta F_{\mathrm{str}}^{\beta }$, given in the table, are reached.