Ising-Like Model for Protein Mechanical Unfolding

The mechanical unfolding of proteins is studied by extending the Wako-Saitô-Muñoz-Eaton model. This model is generalized by including an external force, and its thermodynamics turns out to be exactly solvable. We consider two molecules, the 27th immunoglobulin domain of titin and protein PIN1. We determine equilibrium force-extension curves for the titin and study the mechanical unfolding of this molecule, finding good agreement with experiments. By using an extended form of the Jarzynski equality, we compute the free energy landscape of the PIN1 as a function of the molecule length.

Figure 1

Root mean square length of the 1TIT molecule as a function of the external force $f$ for different temperatures. Inset: cartoon of the model protein under loading.

Figure 2

Average unfolding time of the 1TIT molecule as a function of the force for three values of the temperature. The lines are linear fits of the data to the Arrhenius law discussed in the text. From such fits we find the following values of the unfolding length (in $\AA$): $\tilde{T}=4$, $x_u=3.4\pm 0.1$; $\tilde{T}=6$, $x_u=3.2\pm 0.1$; $\tilde{T}=8$, $x_u=3.0\pm 0.1$.

Figure 3

Plot of the breaking force $f^{*}$ of the 1TIT molecule as a function of the pulling velocity $r$, for the three values of temperature here considered. The lines are fits to the data in the linear regime defined by Eq. (4). From such fits we obtain $x_u=3.1\pm 0.1\AA$ for $\tilde{T}=4$, $x_u=3.0\pm 0.1\AA$ for $\tilde{T}=6$, and $x_u=3.1\pm 0.2\AA$ for $\tilde{T}=8$. Inset: the lines are fits of the data to the equation defining ${\tilde{f}}^{*}$, see text.

Figure 4

Reconstructed free energy landscape $F$ of the PIN1 molecule (lines), as a function of $L$, for different pulling rates $r$ (in $\mathrm{pN}/t_0$ units) and with $\tilde{T}=6$. ●: expected value of $F(L)$ as obtained by direct evaluation of the free energy function. Inset: plot of $F(L)-fL$, for $f=2ϵ/\AA$. The new minimum at $L\simeq 120\AA$ corresponds to the length of the molecule in the large force regime (data not shown).