## 4 Discussion  
Within the classical version of the present MD simulation, performed at the level of single atoms and at femto-second resolution, we have found a clear difference to predictions based on general Poisson statistics. Translocation of ions occurs in single bursts, separated by extended periods of no conduction that can extend over intervals of hundreds of ns (thus extending to the microsecond time scale). This highly irregular behavior can be found to be substantially caused by the coordination energy and barrier height imposed by lone pair charges of carbonyl oxygen atoms in accord with previous findings [2; 3; 4; 6; 8; 9; 10; 26]. A K\({}^{+}\) ion can escape from these 'oxygen cages' if thermal fluctuations and oxygen vibrational modes occasionally exceed certain limits. This perturbation, in turn, exerts an agitation effect on all atoms of the selectivity filter and initiates a cascade of events, involving transitions of ions and water within the filter and new particles entering the filter domain from the core cavity. This later process then continues until the large vibrational amplitudes of the oxygen atoms outside by transferring energy into amino acid backbone atoms and K\({}^{+}\) ions can be captured again by the coordinating forces. The quantum mechanical description of K\({}^{+}\) ion propagation indicates a different situation. This difference becomes especially noticeable if the ion configuration in the filter for a particular site is associated with a vacancy in the neighboring site directed towards the filters exit, e.g., between sites S1 and S0. In these situations, the coordination by the eight surrounding oxygen atoms is slightly weaker because the wave packet of the K\({}^{+}\) ion takes up more volume compared with a classical K\({}^{+}\) ion. This also applies for a wave packet at a minimum uncertainty. As the wave packet further evolves, it starts to spread into the unoccupied adjacent site by at least two mechanisms. One way is provided by 'normal leakage' exerted by high-momentum components present in any localized quantum particle, and involving energies higher than the separating barrier. The other possibility is quantum mechanical tunneling, whose non-negligible influence has recently been emphasized [24]. At this stage it is difficult to make predictions about the relative importance that both mechanisms will contribute to the QM phenomenon that we report. Whatever way may dominate, both processes  
Figure 6: Configuration in the selectivity filter for time t = 0, when the evolution was started, and for time t = 0.227 ps, which is a typical coherence time for this configuration (see Figure 4). In each picture the classical case is on the left, the case with the topmost K\({}^{+}\) ion calculated as a quantum mechanical wave packet in the middle, and the quantum mechanical probability distribution of the ion on the right. (The quantum mechanically calculated K\({}^{+}\) is shown as partly overlapping semi-transparent spheres at different positions and the transparency scales inversely with the probability density at that position.) At t = 0 ps, the quantum mechanical ion has a certain spread due to the Heisenberg uncertainty. At t = 0.227 ps, this wave packet has gained a noticeable extension into the site S0. (An animation showing the entire evolution is available as Supplementary Material). lead to a small but functionally significant presence of the particle's wave function in neighboring binding locations within one (or less) picosecond [7]. The extent to which this early probabilistic presence in the neighboring site leads to a real conduction event depends on the coherence time of the wave packet. Within a dissipative environment the wave packet will be re-created as a new and narrowly localized wave packet at coherence time scales. The place of this localization happens randomly, but with weights corresponding to the probability distribution at this moment. If coherence time is very short, there is little time for the wave packet to spread.