## 2 Materials and Methods
### Quantum Mechanical Simulations  
In quantum mechanical (QM) simulations, one of the K\({}^{+}\) ions evolves as a QM wave packet, whereas other particles continue to be treated in a classical way, with forces given by Equations (1) and (2). The action of the classical particles on the quantum particle is described by an appropriate potential. The reverse action however, exerted by a quantum particle on a classical particle, requires modification of Equations (1) and (2). This modification reflects the typical quantum effect of a dispersion derived from a probability distribution as it emerges from Schrodinger equation. If we denote the position of the quantum mechanical K\({}^{+}\) ion by the vector \(r\), and its charge by \(q\), then integrating the force as given by (1) and (2) provides the potential induced by a classical particle at \(r_{B}\):  
\[V(r,r_{B})=\frac{1}{4\pi\varepsilon_{0}}\frac{qq_{B}}{|r-r_{B}|}\ \ \ \text{if}\ |r-r_{B}|\geq d_{AB} \tag{3}\]  
and  
\[V(r,r_{B})=f_{r\eta\eta A\cdot}(d_{AB}-|r-r_{B}|)\ \ \ \text{if}\ |r-r_{B}|<d_{AB}, \tag{4}\]  
where \(d_{AB}\) is defined as in (1) and (2). Now, the Schrodinger equation for the ion can be written as  
\[\left[-\frac{\hbar^{2}}{2m_{K}}\frac{d^{2}}{dr^{2}}+\sum_{j}V\left(r,r_{j}(t) \right)+V_{Im}(r)\right]\psi(r,t)=ik\frac{\partial}{\partial t}\psi(r,t). \tag{5}\]  
In (5), \(m_{K}\) denotes the mass of the K\({}^{+}\) ion, and the summation '\(j\)' is carried out over all classical particles. The potential \(V_{Im}(r)\) accounts for the overall linear gradient caused by the imposed transmembrane potential at the location of the channel. Solving the above equation is challenged by the fact that the wave function representing the position distribution of the ion \(\left|\psi(r,t)\right|^{2}\) acts back on all classical particles modifying their positions \(r_{j}(t)\). As a result, \(V\left(r,r_{j}(t)\right)\), at a given time \(t\), is determined by \(\psi(r,t)\) over the entire position space. This effect introduces a more difficult non-linearity into the above model. We find, however, that the phase of \(\psi(r,t)\) changes considerably faster as compared to \(V\left(r,\ r_{j}(t)\right)\), and under this condition Equation (5) can be solved in a stepwise manner. During each step, classical particles are 'frozen' for short intervals \(\tau_{cl}\), and during these intervals the quantum mechanical evolution can proceed in the usual linear way. For this linear evolution we employed the numerical Crank Nicolson method [17]. After each interval \(\tau_{cl}\) the resulting position distribution \(\left|\psi(r,t)\right|^{2}\) serves to determine the new positions \(r_{j}\) that can be expected during the next interval \(\tau_{cl}\) due to the action of classical forces proportional to \(\left|\psi(r,t)\right|^{2}\). The resulting potential \(V\left(r,r_{j}(t)\right)\) is finally inserted into Equation (5) to calculate the evolution of \(\psi(r,t)\) during the next interval. The interval \(\tau_{cl}\) is equivalent to a Verlet time step of purely classical MD simulations (usually at 0.25 fs), whereas the time step for the numerical solution of (5) by the Crank Nicolson method was kept to \(\tau_{cl}/100\). It seems to be important to understand the effect of back action of the wave function on the position of classical particles. This requires an additional inspection of the engaged forces. Equations (1) and (2) have to be modified to provide the force between a point particle and a dispersed QM K\({}^{+}\) ion, requiring that integration has to be extended over the entire domain of the dispersed ion.