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Journal of the Chemical Society Faraday Transactions

Quantum conditions in membrane transport controlling nervous excitation

Gerhard Dickel

Published Jan 1, 1991• 0 citations
Publisher

Abstract

While the electric field in synthetic ion-exchange membranes has been found to be described by the equations of Teorell, and Meyer and Sievers, in the case of isotonic osmosis, Planck–Goldman's potential, first found in the nerve, is important. In contrast to Teorell's equation the latter integral represents a potential function. Beneath this integral with respect to position we have to explore the integral with respect to time as results from nervous excitation. This assumes a minimum value if a potential controls its course. Indeed, this integral, the action, yields Planck's constant.To explain this phenomenon, membranes must be considered as polymers doped with Na and/or K atoms. While this process results in the capture of the valence electrons of the dopants by any traps of the membrane, the interaction of these electrons with the remaining cations continues. In this way, the membrane merely represents the carrier of an atomic process resulting in the diffusion of the cations from trap to trap.According to potential theory we have to regard the K˙ and Na˙ potentials in nervous excitation as stationary while the transition from the first state into the second state will be controlled by the Green function. Applying to the partial differential equation of the action controlling Green's function, Jacobi's method of the complete integral we come to the goal if the minimum value of the variational problem in space and time is determined by using Dirichlett's integral. From this we find that in a doped polymer the friction controlling the ion transport depends only on the electrons of the dopant atoms.Finally, from the experimentally determined values of the frictional coefficients one obtains a value of the action which is agreement with Planck's constant to within 6%. As our calculation is based only on classical methods and no quantum conditions were taken into account this result is surprising.

DOI
10.1039/ft9918702093

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While the electric field in synthetic ion-exchange membranes has been found to be described by the equations of Teorell, and Meyer and Sievers, in the case of isotonic osmosis, Planck–Goldman's potential, first found in the nerve, is important. In contrast to Teorell's equation the latter integral represents a potential function. Beneath this integral with respect to position we have to explore the integral with respect to time as results from nervous excitation. This assumes a minimum value if a potential controls its course. Indeed, this integral, the action, yields Planck's constant.To explain this phenomenon, membranes must be considered as polymers doped with Na and/or K atoms. While this process results in the capture of the valence electrons of the dopants by any traps of the membrane, the interaction of these electrons with the remaining cations continues. In this way, the membrane merely represents the carrier of an atomic process resulting in the diffusion of the cations from trap to trap.According to potential theory we have to regard the K˙ and Na˙ potentials in nervous excitation as stationary while the transition from the first state into the second state will be controlled by the Green function. Applying to the partial differential equation of the action controlling Green's function, Jacobi's method of the complete integral we come to the goal if the minimum value of the variational problem in space and time is determined by using Dirichlett's integral. From this we find that in a doped polymer the friction controlling the ion transport depends only on the electrons of the dopant atoms.Finally, from the experimentally determined values of the frictional coefficients one obtains a value of the action which is agreement with Planck's constant to within 6%. As our calculation is based only on classical methods and no quantum conditions were taken into account this result is surprising.
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It is known that the opening or closing mechanism of a voltage-gated ion channel is triggered by the potential difference across the cell membrane in the nervous system. Based on this picture, we model the ion channel as a nanoscale two-terminal ionic tunneling junction. We apply an external time-varying voltage on the junction to mimic the membrane potential difference in the stimulation of neurons. We derive the non-Markovian quantum Langevin equation from the first principle of quantum mechanics for the ion transport in ion channels, and obtain the ion transport current in terms of quantum tunnelings of ions controlled by the time-varying voltage. We find that the time-varying voltage induces an effective magnetic flux, which causes quantum coherence in ion tunnelings. This effective magnetic flux induces further an oscillatory component to the spectral structure of the ion system, forming a time-dependent quantum memory. Such voltage-induced memory is defined as the active quantum memory, which manifests in the system with a regular oscillatory sideband structure in the ion transport current. The sideband structure demonstrates a multicrossing hysteresis in the current-voltage ( I − V ) curve, responding to the variation of the time-varying voltage (membrane potential difference). We also find that the strength of active quantum memory can be described by the ratio of amplitude and frequency of the time-varying voltage, and can be quantitatively measured through the number of nonzero cross points in the I − V hysteresis and conductance-voltage diagram. Additionally, we explore the temperature dependence of the active quantum memory in such a system. Further, we apply this model to the ion channel system on the biological energy scale. The description of these active quantum memory characteristics provides a quantum mechanical understanding to the underlying mechanism of ion channel dynamics.
Hamilton-Jacobi partial differential equations (HJ PDEs) arise in many scientific fields and applications, especially in mechanics and optimal control. Solutions to second order Hamilton-Jacobi partial differential equations (HJ PDEs) have controlled diffusion process representations. Of particular interest is the diffusion representation of an action functional associated with the solution of Schrödinger initial value problems (IVPs). In existing work that connects stochastic control problems to Schrödinger IVPs, one searches for the minimum of an action functional which is the payoff of the control problem. Issues arise, however, as the action functional loses convexity over longer time durations. The time duration where such representation is valid is in fact infinitesimal when the system dimension is infinite. In this dissertation, we present an approach inspired by the Principle of Stationary Action in physics that removes this limitation, which leads to exceptional computational benefits. Instead of searching for local minima of the action functional as in traditional optimal control problems, we look for its stationary values instead. We introduce the ``staticization" operator and stationary-action control problems. A stationary-action stochastic control representation for the dequantized Schrödinger IVP in a non-inertial frame where the potential field is a polynomial is given. A solution approximation as a series expansion in a small parameter is obtained through the use of complex-valued diffusion-process representations. Following this, the staticization operator is studied in detail. A new approach to solving conservative dynamical systems (e.g. paths of objects in gravitational or Coulomb potential field) using stationary-action control-theoretic methods with promising computational benefits is introduced. Two examples of application, the N-body problem and the Schrödinger IVP, are given. Lastly, we demonstrate the existence of strong solutions of a class of degenerate stochastic differential equations (SDEs) that arises in the stationary-action stochastic control representation of Schrödinger IVPs with the Coulomb potential, which is non-smooth and has branch cuts in the complex range. The SDEs we consider have drift terms that have discontinuities and singularities along some manifolds, and diffusion coefficients that are degenerate, and there is no previously existing results regarding the existence of strong solutions for such SDEs.
Mathematical modeling of ionic diffusion along K ion channels indicates that such diffusion is oscillatory, at the weak non-Markovian limit. This finding leads us to derive a Schrödinger-Langevin equation for this kind of system within the framework of stochastic quantization. The Planck's constant is shown to be relevant to the Lagrangian action at the level of a single ion channel. This sheds new light on the issue of applicability of quantum formalism to ion channel dynamics and to the physical constraints of the selectivity filter.
Applying the momentum method, the Fokker-Planck equation is derived from the Boltzmann equation for the r -4 potential between atoms and ions. Two earlier attempts are revealed to be limiting cases of our more general equation, namely for very large and very small mass ratio of the collision partners. The Fokker-Planck equation, after being completed with a decay term and a source term, is solved by the Green's function method. The solution turns out to be composed of modes in the velocity space which decay independently towards a stationary spectrum. The amplitudes of the latter depend on the source term which we assume to be shaped like a delta -function if laser pumping is accounted for. Only for very long relaxation times does the lowest mode prevail in the spectrum which then merges into the Maxwell distribution. Several new lines for ion-atom pairs as well as atom-atom pairs have been measured. They are presented along with the fits using the aforementioned formula. As expected, the agreement is satisfactory only in the first case. In the second case the Fokker-Planck approximation no longer applies.