Influence of molecular variations of ionophore and lipid on the selective ion permeability of membranes: II. A theoretical model

Summary The steady-state electrical properties induced by neutral carriers of ions in lipid bilayer membranes and the time dependence of the membrane current for low applied voltages are described theoretically in terms of a model which allows for a voltage dependence of the interfacial reactions, as well as for a trapezoidal shape of the internal free energy barrier for translocation of the complex. The basic features of the model are closely related to those of others presented previously (J.E. Hall, C.A. Mead & G. Szabo, 1973,J. Membrane Biol. 11:75; S.B. Hladky, 1974,Biochim. Biophys. Acta 352:71; S.B. Hladky, 1975,Biochim. Biophys. Acta 375:327; Eisenman, Krasne & Ciani, 1975,Ann. N.Y. Acad. Sci. 264:34), but the analysis of its consequences on the steady-state and nonsteady-state electrical characteristics is given here in greater detail and is extended to provide the expression for the zero-current potential in ionic gradients. It is shown that parameters, such as the width of the trapezoidal barrier, the plane of the reaction and the ratio of the rate constant of translocation across the membrane interior to the rate constant of dissociation of the complex, can be deduced from steady-state analysis, whereas the individual values of these constants and the distance between the equilibrium positions of the complexes are deducible from relaxation measurements.Definition of the Symbols A _s ^* rate constant for translocation of the neutral carrier across the membrane interior - A _is ^* () defined by Eq. (18) - _is ^* defined by Eq. (24) - B defined by Eq. (9) - c _i , c _i aqueous concentrations of the ionic speciesi in the two bulk solutions - c _s ,c _s ^T ,c _s (0),c _s (d) concentrations of the neutral carrier in the bulk aqueous phases, in the membrane-surrounding torus, and at the ends of the unstirred layers near the membrane-solution interfaces - d membrane thickness - D _s diffusion coefficient of the carrier in the aqueous phase - D _is ^* diffusion coefficient of the complex in the membrane - E _A ,E _B ,E _C electric fields in the compartments shown in Fig. 2 - G(0) conductance near zero voltage - G() conductance at the normalized voltage - I electric current density - J _is flux of complexes across the membrane interior - k _s ^F ,k _s ^B rate constants for the transfer of neutral carriers across the interfaces - k _s ^TM ,k _s ^MT rate constants for the transfer of carriers from the torus into the membrane and vice versa - rate constants of the heterogeneous reaction describing the formation and the dissociation of the ion-carrier complexes - . - L _i () defined by Eq. (26) - N _i defined in Eq. (45) - N _s ^* (1),N _s ^* (2) surface densities of the neutral carrier at their equilibrium positions inside the membrane; note that the equilibrium positions for the neutral carrier, (1) and (2), do not coincide necessarily with the equilibrium positions, (1) _i , and (2) _i , of the complexis. - N _s ^* (st.) defined by Eq. (8) - N _is ^* (1) _i ,N _is ^* (2) _i surface densities of the ion-carrier complexes at their equilibrium positions inside the membrane - q, r fractions of membrane thickness defined in Fig. 1 - V, V ₀ transmembrane potential and potential at zero-current, respectively - defined by Eq. (35) - W _is ^* (x) defined by Eq. (14) - W _i free energy difference between the base and the top of the trapezoid in Fig. 1 - _i width of the flat top of the energy barrier, measured in membrane thickness units - _i distance of the interfacial peaks from the middle of the membrane, measured in membrane thickness units - distance between the two internal free energy wells for the complexes, measured in membrane thickness units (see Fig. 2) - relaxation amplitude - thickness of the unstirred layers - dielectric constant of the membrane phase - _is ^0* (x) standard chemical potential of the ion-carrier complex inside the membrane - transmembrane potential inRT/zF units, namelyzFV/RT=zF(V-V)/RT - (1) _i , (2) _i electric potential at the positions (1) _i , and (2) _i , respectively - ₀ membrane potential at zero current - , net charge of the diffuse double layers per unit membrane area. For small Debye lengths this charge can be viewed as distributed at the membrane-solution interfaces - ₁, ₂ surface charge due to the complexes located at their equilibrium positions - relaxation time - _i defined in Eq. (44)