  Experiments measuring currents through single protein channels show unstable
currents. Channels switch between 'open' or 'closed' states in a spontaneous
stochastic process called gating. Currents are either (nearly) zero or at a
definite level, characteristic of each type of protein, independent of time,
once the channel is open. The steady state Poisson-Nernst-Planck equations with
steric effects (PNP-steric equations) describe steady current through the open
channel quite well, in a wide variety of conditions. Here we study the
existence of multiple solutions of steady state PNP-steric equations to see if
they themselves, without modification or augmentation, can describe two levels
of current. We prove that there are two steady state solutions of PNP-steric
equations for (a) three types of ion species (two types of cations and one type
of anion) with a positive constant permanent charge, and (b) four types of ion
species (two types of cations and their counter-ions) with a constant permanent
charge but no sign condition. The excess currents (due to steric effects)
associated with these two steady state solutions are derived and expressed as
two distinct formulas. Our results indicate that PNP-steric equations may
become a useful model to study spontaneous gating of ion channels. Spontaneous
gating is thought to involve small structural changes in the channel protein
that perhaps produce large changes in the profiles of free energy that
determine ion flow. Gating is known to be modulated by external structures.
Both can be included in future extensions of our present analysis.
