Free-energy landscape of mechanically unfolded model proteins: Extended Jarzinsky versus inherent structure reconstruction

The equilibrium free-energy landscape of off-lattice model heteropolymers as a function of an internal coordinate, namely the end-to-end distance, is reconstructed from out-of-equilibrium steered molecular dynamics data. This task is accomplished via two independent methods: By employing an extended version of the Jarzynski equality and the inherent structure formalism. A comparison of the free energies estimated with these two schemes with equilibrium results obtained via the umbrella sampling technique reveals a good quantitative agreement among all the approaches in a range of temperatures around the “folding transition” for the two examined sequences. In particular, for the sequence with good foldability properties, the mechanically induced structural transitions can be related to thermodynamical aspects of folding. Moreover, for the same sequence the knowledge of the landscape profile allows for a good estimation of the lifetimes of the native configuration for temperatures ranging from the folding to the collapse temperature. For the random sequence, mechanical and thermal unfolding appear to follow different paths along the landscape.

Figure 1

(Color online) Dihedral angle potential, $V_3$, when two or more beads among the four defining $\phi$ are neutral (red dashed curve), and in all of the other cases (blue solid curve). We fixed ${\epsilon }_h=1$ and $S\left({\theta }_i\right)=S\left({\theta }_{i+1}\right)\equiv 0$.

Figure 2

(Color online) Heat capacity $C$ as a function of the temperature $T$ for good (a) and bad (b) folder; the vertical (red) dotted line indicates the hydrophobic collapse temperature $T_{\theta }$ and the horizontal (black) dashed line the value $C_{\mathrm{sol}}$.

Figure 3

(Color online) Probability $P_{\mathit{nc}}$ as a function of the temperature $T$ for good (a) and bad (b) folder; the vertical (magenta) dashed line indicates the folding temperature $T_f$, while the horizontal (black) dotted line refers to the value 0.5.

Figure 4

(Color online) Total energy $⟨E⟩$ as a function of the temperature $T$ for good (a) and bad (b) folder; the solid (red) line corresponds to US’s and the (blue) symbols to FS’s. In the inset an enlargement for low temperatures: The dashed lines indicate the glassy $\left(T_g\right)$ (magenta) and folding $\left(T_f\right)$ (green) temperatures.

Figure 5

(Color online) (a) Free-energy profiles $f_J$ for the good folder as a function of the end-to-end distance $\zeta$ at $T=0.3$, obtained with the EJE for various pulling velocities: From top to bottom $v_p=5\times {10}^{-2}$, $1\times {10}^{-2}$, $5\times {10}^{-3}$, $5\times {10}^{-4}$, $2\times {10}^{-4}$, $2\times {10}^{-5}$, and $5\times {10}^{-6}$. In (b) an enlargement of the curve for $v_p=5\times {10}^{-6}$ at low $\zeta$ is reported. The (black) dashed curve in (a) and (b) refers to the WHAM reconstruction $f_W\left(\zeta \right)$ with $k_W=10$. The number of different pulling experiments performed to estimate the profiles ranges between 150 and 250 at the higher velocities to 28 at the lowest velocity $v_p=5\times {10}^{-6}$. The letters indicate the value of $f\left(\zeta \right)$ for the pulled configurations reported in Fig. 6a and the (blue) vertical solid lines the location of the STs.

Figure 6

(Color online) (a) Pulled configurations of the good folder at $T=0.3$: the NC (a) has ${\zeta }_0\sim 1.9$; the others are characterized by $\zeta =6.8$ (b), $\zeta =16.8$ (c), and $\zeta =27.1$ (d). The beads of type $N$, $B$, and $P$ are colored in green, red and yellow, respectively. (b) Potential energies contributions as a function of the end-to-end distance $\zeta$ estimated during a pulling experiment with speed $v_p=5\times {10}^{-6}$ and obtained by averaging over 28 different realizations at $T=0.3$. (Black) Stars indicate the entire potential energy $V$, (orange) crosses $V_1$, (blue) triangles $V_2$, (magenta) diamonds $V_3$, and (red) squares $V_4$. The (blue) vertical solid lines indicate the transitions previously discussed in the text.

Figure 7

(Color online) Free-energy profiles $f_J\left(\zeta \right)$ obtained with the EJE for good folder at various temperatures: $T=0.2$ (red squares), 0.4 (green stars), 0.5 (orange circles), 0.6 (magenta plus), and 0.7 (blue triangles). In the inset, an enlargement is reported at small $\zeta$. Data refer to $v_p=5\times {10}^{-4}$. The number of different realizations performed to estimate the averages at the different temperatures ranges between 160 and 250.

Figure 8

(Color online) Average unfolding times $\tau$ for the $GF$ at various temperatures corresponding to ${\zeta }_{\mathrm{th}}=4$. Filled (black) circles denote the numerical data, the estimations obtained via Eq. (24) and Eq. (25) are represented by empty (blue) diamonds and (red) stars, respectively. The arbitrary scaling factor entering in Eq. (25) (see text) has been set equal to 8. The average times have been estimated over 100 000–200 000 unfolding events for $T=0.7$ and 0.6, 12 000 events at $T=0.5$ and as few as 200 and 60 events at the lowest temperatures, namely $T=0.4$ and 0.3.

Figure 9

(Color online) (a) Free-energy profiles $f_J$ for the bad folder as a function of the end-to-end distance $\zeta$, obtained with the EJE for various pulling velocities: From top to bottom $v_p=5\times {10}^{-4}$ and 160 realizations (black circles), $2\times {10}^{-4}$ and 200 realizations (red squares), $1\times {10}^{-4}$ and 200 realizations (blue triangles), $5\times {10}^{-5}$ and 100 realizations (green diamonds), $5\times {10}^{-6}$ and 28 realizations (magenta stars). The WHAM estimate $f_W\left(\zeta \right)$ is also shown (black dashed line). In the inset an enlargement of the curve at low $\zeta$ for $v_p=5\times {10}^{-6}$ is reported together with $f_W\left(\zeta \right)$. Data have been obtained at $T=0.3$. (b) Potential energies contribution as a function of the end-to-end distance $\zeta$ estimated during a pulling experiment with velocity $v_p=5\times {10}^{-6}$ and obtained by averaging over 28 different realizations at $T=0.3$. Black stars indicate the entire potential energy $V$, (orange) crosses $V_1$, (blue) triangles $V_2$, (magenta) diamonds $V_3$, and (red) squares $V_4$. The (blue) solid lines indicate the transitions discussed in the text.

Figure 10

(Color online) Pulled configurations of the bad folder at $T=0.3$: The reported configurations refer to ${\zeta }_0=4.7$ (NC) (a), $\zeta =9.9$ (b), 14.5 (c), 22.1 (d), 24.6 (e), and 29.7 (f).

Figure 11

(Color online) Free-energy profiles $f_J\left(\zeta \right)$ obtained via the EJE for bad folder at three temperatures: Namely, $T=0.2$ (red squares), $T=0.3$ (orange circles), $T=0.4$ (blue stars). In the inset an enlargement is reported at small $\zeta$. Data refer to pulling velocity $v_p=5\times {10}^{-6}$ and the averages are performed over 28 samples of the same protocol.

Figure 12

(Color online) Free-energy profiles $f_J$ (blue solid lines) as a function of $\zeta$ for various temperatures for the good folder: (a) $T=0.3$ for $v_p=5\times {10}^{-6}$ and 28 repetitions; (b) $T=0.4$ for $v_p=5\times {10}^{-4}$ and 240 experiments; (c) $T=0.5$ for $v_p=5\times {10}^{-4}$ and 240 repetitions. The (black) dashed lines refer to the WHAM estimation $f_W\left(\zeta \right)$, (green) squares to $f_{\mathrm{IS}}\left(\zeta \right)$ obtained by employing the TDB and (red) circles to $f_{\mathrm{IS}}\left(\zeta \right)$ obtained by employing the ISs in the PDB for each considered $T$.

Figure 13

(Color online) Free-energy profiles $f_J$ as a function of $\zeta$ for various temperatures for the bad folder: (a) $T=0.3$ and (b) $T=0.4$. The data refer to a pulling velocity $v_P=5\times {10}^{-6}$ and 28 repetitions of the same pulling protocol. The symbols are the same as in Fig. 12.

Figure 14

(Color online) Reconstructed $V_{\mathrm{IS}}\left(\zeta \right)$ (lower panel) and $R_{\mathrm{IS}}\left(\zeta \right)$ (upper panel) for good folder (a) and bad folder (b) by employing ISs in the PDB at $T=0.3$. In the lower panel the blue dotted line refers to the average potential energy evaluated during the corresponding pulling experiments [this has been already reported in Fig. 6b for the $GF$ and in Fig. 9b] for the $BF$. Please notice that the data have been vertically translated in order to have zero energy at the NC.

Figure 15

(Color online) End-to-end distance of the ISs estimated during USs at various temperatures: (Black) Circles represent the average value; (blue) stars the minimal value; and (red) squares the maximal one. The upper panel refers to the $BF$ and the lower one to the $GF$. The horizontal magenta dashed line indicates the ${\zeta }_{\mathit{\text{trans}}}$ value. For the $GF$ (respectively, $BF$) trajectories of duration $t\sim 100000–500000$ (respectively, $t\sim 50000–250000$) have been examined to obtain the ISs at constant time intervals $\Delta t=5$.