# Slow conductance changes due to potassium depletion in the transverse tubules of frog muscle fibers during hyperpolarizing pulses

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## Summary

When hyperpolarizing currents are applied between the inside and outside of a muscle fiber it is known that there is a slow transient decrease (300- to 600-msec time constant) in the measured fiber conductance sometimes referred to as “creep” which is maximal in K_{2}SO_{4} Ringer's solutions and which disappears on disruption of the transverse tubular system. An approximate mathematical analysis of the situation indicates that these large, slow conductance changes are to be expected from changes in the K^{+} concentration in the tubular system and are due to differences in transport numbers between the walls and lumen of the tubules. Experiments using small constant-voltage and constant-current pulses (membrane p. d. changes ≲20 to 30 mV) on the same fibers followed by an approximate mathematical and more exact computed numerical analysis using the measured fiber parameters and published values of tubular system geometry factors showed close agreement between the conductance creep predicted and that observed, thus dispensing with the need for postulated changes in individual membrane conductances at least during small voltage pulses. It is further suggested that an examination of creep with constant-voltage and constant-current pulses may provide a useful tool for monitoring changes in tubular system parameters, such as those occurring during its disruption by presoaking the fibers in glycerol.

## Keywords

K2SO4 Membrane Conductance Transport Number System Geometry Fiber Conductance## Table of main symbols used

*R, T, F*Gas constant, Temperature in °K and the Faraday

*a*Fiber radius

*r*Radial distance from the center of the fiber (

*cf.*Fig. 2*A*)*t*Time in sec

*V*_{1},*V*_{2}Voltages measured by electrodes 1 and 2 (

*cf.*p. 248)- λ
Longitudinal fiber space constant (

*λ*^{2}=*R*_{ m }*a*/2*R*_{ i })*R*_{m},*R*_{m}*(t)*Total membrane resistance per unit surface area of fiber (Ω cm

^{2})*R*_{m}(0),*R*_{m}(∞)As above at

*t*=0 (excluding capacity transient) and at*t*=∞ during a current or voltage pulse*G*_{m},*G*_{m}*(t)*Total membrane conductance (mho·cm

^{−2}) per unit area of fiber surface*G*_{m}(0),*G*_{m}(∞)As above at

*t*=0 (excluding the capacity transient) and at*t*=∞ during a current or voltage pulse*R*_{sm},*G*_{sm}Surface membrane resistance (Ω cm

^{2}) and conductance (mho·cm^{−2}), respectively, excluding the TTS*R*_{T},*G*_{T}Input resistance (Ω cm

^{2}) and conductance (mho·cm^{−2}) of the TTS referred to unit area of fiber surface*f*_{T}Fraction of the K

^{+}conductance in the TTS to the total K^{+}conductance of the fiber [*cf.*Eq. (7)]*R*_{i}Internal resistivity of the fiber (Ω cm)

*r*_{s}Electrical access resistance of the TTS [Ω cm

^{2};*cf.*Fig. 3 and Eq. (24)]*h*Diffusional access resistance of the TTS [

*cf.*Eq. (27)]*I*_{0}Total current entering fiber (amp)

*I*_{m},*i*_{m}Total current per unit area of fiber surface (amp·cm

^{−2}; considered positive in the hyperpolarizing direction)*i*_{sm}Current going through the surface membrane alone (amp·cm

^{−2};*cf.*Fig. 3)*i*_{0},*i*_{0}(*t*)Total current entering the TTS referred to unit area of surface membrane (amp·cm

^{−2};*cf.*Fig. 3)*I*_{K},*I*_{K}(*r*)K

^{+}current density crossing the equivalent TTS disc at radial distance*r*[*cf.*Fig. 2*A*and Eq. (23)]*i, i(r, t)*Radial current in the lumen of the TTS at radial distance

*r*and time*t*(*cf.*Fig. 2*B*)*C, C(r, t)*K

^{+}concentration within the TTS at radial distance*r*and time*t*(mEquiv·liter^{−1})*C*_{o},*C*_{K}Both refer to external solution and initial TTS K

^{+}concentration (mEquiv·liter^{−1})*V, V(r, t)*The potential at radial distance

*r*in the lumen of the TTS with respect to the external solution at time*t*(*cf.*Figs. 2 and 3)*V(a), V(a, t)*The p.d. across the access resistance (

*cf.*Figs. 3*B*and 3*C*)*V*_{0},*V*_{0}(*t*)The potential of the sarcoplasm with respect to the external solution (

*cf.*Figs. 2 and 3)*E*_{K}The K

^{+}equilibrium potential between the sarcoplasm and the externa solution or across the tubular wall*t*_{K}^{m},*t*_{K}^{s}The transport number for K

^{+}in the TTS membranes and in the solution of the tubular lumen, respectively- ρ
The fraction of fiber volume occupied by tubules, and not implicitly including branches

- ρ′
As above but always including branches

- σ
A dimensionless network factor for the TTS

*G*_{W}Conductance per unit area of tubular wall (mho·cm

^{−2})*G*_{L}Conductance of tubular lumen (mho·cm

^{−1})- ξ
Volume-to-surface ratio of the TTS

- \(\bar G_W \)
Effective wall conductance of TTS membranes per unit volume of fiber [mho·cm

^{−3};*cf.*Eq. (14)]- \(\bar G_L \)
Effective radial conductance of the lumen of the TTS per unit volume of fiber [

*cf.*Eq. (20)]*d*The thickness of the equivalent disc representing the TTS [

*cf.*Eq. (15)]- λ
_{T} Space constant of the TTS [

*cf.*Eq. (37).*cp.*Eq. (11)]*D*_{K}The diffusion coefficient of K

^{+}ions in the lumen of the TTS (cm^{2}sec^{−1})- \(\bar D_K \)
The effective radial K

^{+}diffusion coefficient in the TTS [*cf.*Eq. (28)]*J*_{0},*J*_{1}Bessel functions of order “0” and “1”, respectively

*I*_{0},*I*_{1}Modified Bessel functions of order “0” and “1”, respectively

- τ
Time constants of slow conductance changes

- τ
_{vc} Time constant of slow conductance changes during a constant-voltage pulse

- τ
_{cc} Time constant of slow conductance changes during a constant-current pulse

- α, α
_{m} Roots of various Bessel function equations

*g*_{1},*g*_{2},*g*_{3},*g*_{4}Constants used to fit cubic equation for conductance-voltage curves [

*cf.*Eq. (71)]

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