## 3 Results
### Unbinding of the TFO from the Double Helix
The I\({}_{1}\) states of these switches are characterized by low energy minima in which the bases at the \(5^{\prime}\) end of the TFO are still interacting with the duplex. The Ts\({}_{\text{unb}}\) state, which is characterized by structures with a TFO-DH distance of 23-24 A, and by a complete loss of HBs interactions, is reached through the sampling of a high-energy conformational basin, as observed for the diprotic switch (Figure 3A-C). These observations are confirmed by the per-base unbinding profiles that are similar for all of the three tetraprotic switches, which are characterized by the removal of the base-pair interactions starting from the \(3^{\prime}\) to the \(5^{\prime}\) end of the TFO (Figure 4D-F), independently of the linker length. Figure 2: Iso-energetic contour of the free energy surface plotted as a function of the HB number established between the TFO and the DH, and of the distance between their centers of mass for the DIPRO5 (**A**), DIPRO15 (**B**) and DIPRO25 (**C**) systems. The upper structures are the representative configurations of the most populated clusters of the Bound (B), Intermediate (I\({}_{0}\) and I\({}_{1}\)) and Transition (TS\({}_{\text{unb}}\)) states, extracted from the trajectories. The color scheme follows that described in Figure 1.  
A comparative free energy profile along a minimum energy path, as a function of a generalized reaction coordinate, is plotted in Figure 5A,B for the DIPRO and TETRA switches, respectively. The energy value of the B state for the DIPRO5 switch (Figure 5A red line) is two times lower than those observed for the DIPRO15 and DIPRO25 systems (Figure 5A green and blue lines, respectively). Moreover, the DIPRO5 B and I1 states are separated by a relatively high activation energy (Figure 5A, red line), again almost twice than that required for the TFO unbinding in the DIPRO15 and DIPRO25 systems (Figure 5A, green and blue lines, respectively). In the case of the TETRA5 (Figure 5B, red line), TETRA15 (Figure 5B, blue line) and TETRA25 switches (Figure 5B, green line), the energy values observed for the B state are half of those observed for the diprotic switches, with the I0 and I1 states separated by very low activation energies, which favour the transition towards the unbound states. These data indicate that the length of the linker connecting DH with TFO has a crucial role in stabilizing the DH-TFO interaction only in the presence of two protonation sites. In this case, the DH-TFO interaction energy strongly decreases upon increasing the loop length, indicating that the entropic contribution related to the loop length has a direct effect on the TFO unfolding energy. This effect is not observed for the tetraprotic switches. It is interesting to note that the experimental results indicate a two unit increase in the pKa switching mechanism only for the diprotic system, when the linker length is reduced from 25 to 5 bases [29], in line with the simulative findings presented here. Figure 3: Iso-energetic contour of the free energy surface plotted as a function of the HB number established between the TFO and the DH, and of the distance between their centers of mass for the TETRA5 (**A**), TETRA15 (**B**) and TETRA25 (**C**) systems. The upper structures are the representative configurations of the most populated clusters of the Bound (B). Intermediate (I0 and I1) and Transition (TS\({}_{\text{unb}}\)) states, extracted from the trajectories. The color scheme follows that described in Figure 1.  
Figure 4: Per-base unbinding profile of TFO from the DH as a function of time. For each 2-ns time window, the distance between the centers of mass of the TFO and of the DH C2\({}^{\prime}\) atoms was calculated, and the corresponding rectangle in the plot was colored if the average distance was lower than 3.2 Å. A white rectangle indicates the loss of the interaction. The profile is reported for the DIPRO5 (**A**), DIPRO15 (**B**), DIPRO25 (**C**), TETRA5 (**D**), TETRA15 (**E**) and TETRA25 (**F**) switches.