## 3 Results  
### Unbinding of the TFO from the Double Helix  
The TFO unbinding profile from the double helix was analyzed in terms of its free energy surface (FES) and plotted as a function of the number of hydrogen bonds (HBs) established between the TFO and the DH, and of the distance between their centers of mass, as reported in Figures 2 and 3 for the diprotic and tetraprotic switches, respectively. For all of the six systems, the FES shows the presence of three minima corresponding to the bound state (B) and two intermediate (I\({}_{0}\) and I\({}_{1}\)) states, separated by two transition states (Ts and Ts\({}_{\text{unb}}\)). In particular, the Ts\({}_{\text{unb}}\) state represents the final step of the ABMD simulations (Figure 2A-C or Figure 3A-C). For each analyzed structure, conformations belonging to the B, I and Ts\({}_{\text{unb}}\) states were clustered, and the representative structures are shown in the upper part of Figure 2A-C or Figure 3A-C.  
For the DIPRO5 switch, the B state is stable, being characterized by the presence of 13-14 HBs and a distance between TFO and DH centers of mass close to 2 A (Figure 2A). The I\({}_{0}\) state has a lower number of HBs and an average TFO-DH distance of 6 A (Figure 2A) due to the unbinding of three nucleotides at the 3\({}^{\prime}\) end of the TFO. In the I\({}_{1}\) state, the number of HB interactions is reduced to four, and in the final Ts\({}_{\text{unb}}\) state only two to three HBs are present. In fact, in the Ts\({}_{\text{unb}}\) state, the TFO is not fully detached because the first three bases at its 5\({}^{\prime}\) end are still interacting with the DH. The per-base unbinding profile of the DIPRO5 switch, reported in Figure 4A, provides a clear view of the detaching mechanism described by the FES, showing the progressive loss of HBs from the 3\({}^{\prime}\) to the 5\({}^{\prime}\) end of the TFO. The DIPRO15 system shows the presence of less stable minima than those observed in the DIPRO5 one (Figure 2B). In detail, the B state shows the presence of 8-10 HBs, although the TFO is still quite close to the DH. The I\({}_{0}\) state is similar to that of the DIPRO5 switch, showing the unbinding of the first three nucleotides at the TFO\(3^{\prime}\) end. The I\({}_{1}\) state samples several loose structures, showing a concurrent loss of three base pair interactions at both the \(3^{\prime}\) and \(5^{\prime}\) ends of the TFO (Figure 2B). Starting from this configuration, the Ts\({}_{\text{unb}}\) state is reached by an extensive sampling of a high-energy conformational basin, characterized by a TFO-DH distance of about 20 A and the loss of all HBs (Figure 2B), indicative of a full unbinding of the TFO from the DH. The per-base unbinding profile of the DIPRO15 confirms the removal of the base-pair interactions at both the \(3^{\prime}\)and \(5^{\prime}\) ends of the TFO, followed by the loss of the internal bases' interactions, leading to the complete detachment of the TFO from the DH (Figure 4B). The DIPRO25 switch is the less stable one, being characterized by high-energy minima, with the B state easily evolving towards the I\({}_{0}\) state (Figure 2C). In this case, the unbinding process of the TFO from the DH, as observed through the structures sampled in the I\({}_{0}\) and I\({}_{1}\) states and from the per-base unbinding profile in Figure 4C, follows an opposite path when compared to the DIPRO5 system because the detachment starts from the TFO \(5^{\prime}\) end. The Ts\({}_{\text{unb}}\) state, identified by structures with a TFO-DH distance of 24-25 A and characterized by a complete loss of HBs interactions, is reached through the sampling of a high-energy conformational basin (Figure 2C). For all the tetraprotic switches, the B state is characterized by metastable high energy minima, which evolve towards the I\({}_{0}\) and I\({}_{1}\) states (Figure 3A-C) by sampling several conformations characterized by a rapid loss of HBs.