| Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 11, Iss. 3, Jul, 2007, pp. 293-308 @2007 Society for Chaos Theory in Psychology & Life Sciences Dynamics of Cell Membrane Passive Depolarization: A Phase Portrait Abstract: Does a persistent blockage of the ionic pumps bring cell
membrane voltage to zero? This apparently trivial question of basic
cellular Biology stirred up an intriguing problem of nonlinear
dynamics. A 3-ion model based on continuity and charge conservation
proves that membrane voltage actually sets on a negative value,
meaning that chemical equilibrium is never reached, rather an
inversion of the Na+ concentration gradient occurs, usually hours
after the blockage of the pumps. Experimental tests carried out with
PC12 cells incubated with Oubaine for a period of 24 hours show an
increase of cytosolic Na+ of about 266 mM/l with respect to a control
sample. The result is compatible with an inversion of the Na+ gradient,
which eventually brings the membrane voltage to a negative value.
Reactivation of the Na+-K+ pumps even after a prolonged period of
blockage (late repolarization) should lead to repolarization and
revival of the cell. In the 3D space of the ionic concentrations,
the dynamics of passive depolarization reveals an intriguing topology,
all trajectories being stacked in parallel planes, each set ending to a
unique fixed point via an infinitely dense set of lines. The dynamics of
repolarization has a different phase portrait, especially in the case of
late repolarization. Thus, a sequence of depolarization- repolarization
cycles may result in a path wandering in the phase space, or in a closed
loop, depending on the timing of the sequence. Keywords: passive depolarization, nonlinear dynamics, fixed points, phase portrait, late repolarization |