Painlev\'e structure of a multi-ion electrodiffusion system

A nonlinear coupled system descriptive of multi-ion electrodiffusion is investigated and all parameters for which the system admits a single-valued general solution are isolated. This is achieved \textit{via} a method initiated by Painleve' with the application of a test due to Kowalevski and Gambier. The solutions can be obtained explicitly in terms of Painleve' transcendents or elliptic functions.Comment: 9 p, Latex, to appear, J Phys A FT

Long-range interactions, voltage sensitivity, and ion conduction in S4 segments of excitable channels.

Forces acting on the S4 segments of the channel, the voltage-sensing structures, are analyzed. The conformational change in the Na channel is modeled as a helix-coil transition in the four S4 segments, coupled to the membrane voltage by electrical forces. In the model, repulsions between like charges make the S4 segment unstable, but field-dependent forces hold it in an alpha-helix configuration at resting potential. At threshold depolarization, the S4 helices cooperatively expand into random coils, breaking the hydrogen bonds connecting adjacent loops of the alpha helices. Exposed electron pairs left on the carbonyl oxygens constitute sites at which cations can bind selectively. The first hydrogen bond to break is at the channel exterior, then the second breaks, and so on in a zipper-like motion along the entire segment. The Na+ ions hop from one site to the next until all H bonds are broken and all sites are filled with ions. This completes the pathway over which the permeant ions move through the channel, driven by the electrochemical potential difference across the membrane. This microscopic mechanism is consistent with the thermodynamic explanation of ion-channel gating previously formulated as the ferroelectric-superionic transition hypothesis

Does the Na channel conduct ions through a water-filled pore or a condensed-state pathway?

Many investigators assert that the ion-conducting pathway of the Na channel is a water-filled pore. This assertion must be reevaluated to clear the way for more productive approaches to channel gating. The hypothesis of an aqueous pore leaves the questions of voltage-dependent gating and ion selectivity unexplained because a column of water can neither serve as a switch nor provide the necessary selectivity. The price of believing in an aqueous pore therefore is a futile search for separate ad hoc mechanisms for gating and selectivity. The fallacy is to assume that only water is available to carry ions rapidly, ignoring the role of the glycoprotein, which can form an elastomeric phase with water. The elastomer is a state of matter, neither liquid nor solid, in which the molecules of a liquid are threaded together with cross-linked polymer chains; it supports fast ion motion (Owen, 1989). An alternative hypothesis for channel gating, based on condensed-state materials science, already exists (Leuchtag, 1988, 1991a). The ferroelectric-superionic transition hypothesis (FESITH) postulates that the Na channel exists in a metastable ordered (closed) state at resting potential and, on threshold depolarization, undergoes a reversible order-disorder phase transition to a less-ordered, ion-conducting (open) state. The ordered state is ferroelectric; the disordered state is a fast ion conductor selective for Li+ and Na+. The basis of the voltage dependence is elevation of transition temperature with electric field, well established in ferroelectrics. FESITH is consistent with single-channel transitions, gating currents, heat and cold block, and other phenomena observed at channel or membrane level. An implication of FESITH, the Curie-Weiss law, has been shown to explain existing data on membrane capacitance versus temperature in squid axon (Leuchtag, 1991c). Only on the basis of a clear understanding of function can we expect new structural data on the Na-channel glycoprotein to generate realistic structure-function models at the molecular level

Steady-state electrodiffusion. Scaling, exact solution for ions of one charge, and the phase plane.

This is the first of two papers dealing with electrodiffusion theory (the Nernst-Planck equation coupled with Gauss's law) and its application to the current-voltage behavior of squid axon. New developments in the exact analysis of the steady-state electrodiffusion problem presented here include (a) a scale transformation that connects a given solution to an infinity of other solutions, suggesting the po-sibility of direct comparison of electrical data for membranes with different thicknesses and other properties; (b) a first-integral relation between the electric field and ion densities more general than analogous relations previously reported, and (c) an exact solution for the homovalent system, i.e., a membrane system permeated by various ion species of the same charge. The latter is a generalization of the known one-ion solution. The properties of the homovalent solution are investigated analytically and graphically. In particular we study the phase-plane curves, which reduce to the parabolas discussed by K. S. Cole in the special case in which the current-density parameter (a linear combination of the ionic current densities) is zero

Fluctuation and linear analysis of Na-current kinetics in squid axon.

The power spectrum of current fluctuations and the complex admittance of squid axon were determined in the frequency range 12.5 to 5,000 Hx during membrane voltage clamps to the same potentials in the same axon during internal perfusion with cesium. The complex admittance was determined rapidly and with high resolution by a fast Fourier transform computation of the current response, acquired after a steady state was attained, to a synthesized signal with predetermined spectral characteristics superposed as a continuous, repetitive, small perturbation on step voltage clamps. Linear conduction parameters were estimated directly from admittance data by fitting an admittance model, derived from the linearized Hodgkin-Huxley equations modified by replacing the membrane capacitance with a "constant-phase-angle" capacitance, to the data. The constant phase angle obtained was approximately 80 degrees. At depolarizations the phase of the admittance was 180 degrees, and the real part of the impedance locus was in the left-half complex plane for frequencies below 1 kHz, which indicates a steady-state negative Na conductance. The fits also yielded estimates of the natural frequencies of Na "activation" and "inactivation" processes. By fitting Na-current noise spectra with a double Lorentzian function, a lower and an upper corner frequency were obtained; these were compared with the two natural frequencies determined from admittance analysis at the corresponding potentials. The frequencies from fluctuation analyses ranged from 1.0 to 10.3 times higher than those from linear (admittance) analysis. This discrepancy is consistent with the concept that the fluctuations reflect a nonlinear rate process that cannot be fully characterized by linear perturbation analysis. Comparison of the real part of the admittance and the current noise spectrum shows that the Nyquist relation, which generally applies to equilibrium conductors, does not hold for the Na process in squid axon. The Na-channel conductance, gamma Na, was found to increase monotonically from 0.1 to 4.8 pS for depolarizations up to 50 mV from a holding potential of -60 mV, with no indication of a maximum value