  The mechanism behind the high throughput rate in K$^{+}$ channels is still an
open problem. Recent simulations have shown that the passage of potassium
through the K$^{+}$ channel core, the so-called selectivity filter (SF), is
water-free against models where the strength of Coulomb repulsion freezes ions
conduction. It has been suggested that quantum coherent hopping might be
relevant in mediating ion conduction. Within the quantum approach and the
hypothesis of desolvated ions along the pathway, we start with a number of
particles in a source to see how they go across the SF modeled by a linear
chain of sites to be collected in a drain. As a main result we show that there
is a threshold SF occupancy is three ions on average, which is in agreement
with recent classical model simulations.
