Relationship Between Transmembrane Inorganic Ion Flux and Transmembrane Potential

Membranes surround the individual cells of animals and organelles within the cell. They are composed of lipids and proteins. Specific proteins are responsible for the transport of specific inorganic ions across the membrane. Invariably, this transport of inorganic ions across the cell membrane is accompanied by changes in the transmembrane voltage. The equilibrium transmembrane potential for a specific inorganic ion, for instance K⁺, is given by the Nernst equation:

⇒ Equation [1]

EK+ = RT In [K+]o   ZF [K+]i

R, T, and F are the molar gas constant, absolute temperature, and Faraday constant, respectively. Z represents the valence of the inorganic ion and the subscripts o and i indicate whether the ion is outside or inside the cell membrane. The transmembrane potential of neurons is around −60 mV. In the nervous system, changes in the transmembrane potential due to a change in the flux rate of inorganic ions can be propagated rapidly and over distances as long as several feet via the axon, a long projection of many nerve cells (Fig. 1). At the axonal terminal, the voltage change initiates a process leading to the flux of calcium ions into the nerve terminal. This results in the secretion of chemical signals, neurotransmitters, which bind to membrane-bound proteins, neurotransmitter receptors, on adjacent cells. Upon binding specific neurotransmitters, the receptors transiently open transmembrane channels. The channels are permeable to Na⁺, K⁺, or Cl⁻, depending on the receptor. The resulting changes in the transmembrane voltage may lead to propagation of a signal to an adjacent cell. Thus, this interplay between chemical reactions and transmembrane voltage changes plays a decisive role in the rapid communication between nerve (and nerve and muscle) cells and in nervous system function.

In 1890, Max Planck derived the relationship between the rate of movement of inorganic cations and anions across a porous barrier and the resulting electric field. If one assumes a constant electric field and constant inorganic ion concentration, the Planck equation is easily integrated and can be used to estimate the transmembrane voltage change, V_m, that results from the flow of inorganic ions across cell membranes. The resulting Goldman equation is

⇒ Equation [2]

Vm = RT In PK(K+)o + PNa(Na+)o + PCl[Cl−]i   F PK(K+)i + PNa(Na+)i + PCl[Cl−]o

P_K, P_Na, and PCl⁻ represent the permeability coefficient of the membrane for K⁺, Na⁺, and Cl⁻, respectively. [K⁺], [Na⁺], [Cl⁻] represent the molar concentrations of the ions, and the subscript o or i indicates whether the ions are outside or inside the cell membrane. As usual, R, T, and F represent the molar gas constant, the absolute temperature, and the Faraday constant respectively.

How is the rate of ion movement through a proteinformed channel across a cell membrane related to the transmembrane voltage? In the nervous system, signal transmission is regulated by the binding of chemical signals, neurotransmitters, to membrane-bound proteins, called receptors. Commonly, when two molecules of a neurotransmitter have bound to the receptor, the protein forms a transmembrane channel that remains open for a few milliseconds, allowing the receptor-specific passage of sodium, potassium, or chloride ions. The chemical reaction for many neurotransmitter receptors can be written as shown in Fig. 2; in this case the kinetic mechanism of the nicotinic acetylcholine receptor is used as an example. This receptor plays an important role in signal transmission between nerve cells in the brain and between nerve and muscle cells (Fig. 1).

Figure 2 Minimum mechanism to account for the rates of a neurotransmitter (acetylcholine) receptor-mediated cation translocation and for receptor inactivation and reactivation as a function of acetylcholine concentration. The active (A) and inactive (I) forms of the receptor bind neurotransmitter (L) in rapidly achieved equilibria denoted by the microscopic equilibrium constants (K). Active receptor with two bound ligand molecules (AL2)[AL bar with the base of 2] converts rapidly (1 to 2 msec) to an open channel (AL2)[AL bar with the base of 2] with an equilibrium constant for channel opening (1/Φ ·Φ=kcl /kop where kop and kcl are the rate constants for channel opening and closing respectively ). AL2[AL bar with the base of 2] permits the movement of inorganic Na+ and K+ ions through the membrane, where Jm, is the observed rate constant for the flux of inorganic ions (Na+, K+) through the open receptor-formed transmembrane channel (see Eq. 3). In the continued presence of neurotransmitter, the receptors reversibly form inactive forms I in the 10- to 200-msec time region, depending on the receptor and the concentration of neurotransmitter. This process is called receptor desensitization. (Reproduced with permission from Cash, D. J., Aoshima, H., and Hess, G. P. (1981) Proc. Natl. Acad. Sci. USA 78, 3381–3322.)

The specific reaction rate for the transmembrane flux of inorganic cations controlled by the nicotinic acetylcholine receptor, J_m[J bar to the base of m], has a value of about 5 × 10⁷ M⁻¹ sec⁻¹ at 14°C. The relationship between the permeability coefficient P for a specific inorganic ion M^± (Eq. 2) and the specific reaction rate J_m[J bar to the base of m] is given by:

⇒ Equation [3]

PM± = JM±R0(. AL 2)

R₀ represents the moles of specific receptors in the cell membrane and (AL₂)[AL bar to the base of 2] the fraction of the receptors that are in the open-channel form. Equations 2 and 3, therefore, establish the important relationship between the receptor controlled receptorcontrolled movement of inorganic ions through the cell membrane and the resulting change in transmembrane voltage, V_m.