J. Physiol. (1979), 293, pp. 301-318
301
With 14 text-ftgures Printed in Great Britain
A VOLTAGE-DEPENDENT GATE IN SERIES WITH THE INWARDLY RECTIFYING POTASSIUM CHANNEL IN FROG STRIATED MUSCLE
BY E. MANCINELLI AND A. PERES From the Istituto di Fisiologia Generale e Chimica Biologica, Universitd di Milano, Via Mangiagalli, 32, Milano, Italy
(Received 25 April 1978) SUMMARY
1. The degree of tubular potassium depletion and the decrease of potassium conductance due to hyperpolarizing pulses in striated muscular fibres have been examined with the three micro-electrode voltage-clamp technique. 2. The conductance of the fibre membrane has been measured in different extracellular K+ concentrations from 1 to 10 mm. 3. Comparison of the two sets of measurements shows that changes in tubular K+ concentration are not sufficient to account for the conductance decrease associated with hyperpolarization. 4. The presence of a voltage-dependent gate in series with the inwardly rectifying channel for K+ ions, suggested by Almers (1972a, b), is thus confirmed. INTRODUCTION
In recent years K+ depletion or accumulation in extracellular spaces has received considerable attention. When the membrane of a striated muscle fibre is hyperpolarized, the inward current declines to reach a steady level (Adrian & Freygang, 1962; Adrian, Chandler & Hodgkin, 1970b). In this preparation, in the range of potentials negative to the resting potential, only the inwardly rectifying channel for K+ seems to be operative. While for hyperpolarizing pulses the instantaneous current voltage relation is reasonably linear, the steady-state relation shows a region of negative slope conductance- for extreme hyperpolarizations. This feature has been already discussed by Adrian et al. (1970b) and by Almers (1972a, b). These authors pointed out that it is difficult to explain the negative conductance simply on the basis of a decrease of potassium concentration in the transverse tubules. Indeed, since the inward rectifier resides mostly in the tubular membrane (Nakajima, Nakajima & Peachey, 1969; Hodgkin & Horowicz, 1960), the passage of inward current would cause a decrease in the tubular K+ concentration (depletion) if the replenishment of the tubules from the bulk solution is limited by a low transport number for potassium. This would make the K+ equilibrium potential across the tubular membrane more negative, leading to a decrease of the driving force V - VK. The tubular potassium current then should decrease with increasing depletion. 0022-3751/79/4200-0423 $01.50 © 1979 The Physiological Society
302 E. MANCINELLI AND A. PERES In the steady state, however, the K+ current leaving the tubules through the membrane must equal the current entering the tubules from the bulk solution. If one makes the reasonable assumption that the current entering is proportional to the concentration difference between bulk solution and the tubular lumen, the steady-state current should increase with increasing depletion. To explain this inconsistency with his experimental results, Almers (1972a, b) suggested, in addition to the tubular depletion, the presence of a voltage-dependent gate in series with the inwardly rectifying channel. This structure, while fully open at the resting potential, tends to close when the potential is made more negative than - 120 mV. In the experiments reported here the existence of this voltage-dependent gate has been examined. We compared the conductance of fibres previously depleted by hyperpolarizing pulses with the conductance of non-depleted fibres immersed in 'different external K+ concentrations. The results show that changes in tubular K+ concentration are insufficient to account for the conductance decline and favour the hypothesis of the existence of a voltage-dependent gate. METHODS Whole semitendinosus muscles of Rana ewculenta were dissected out. The three micro-electrode voltage-clamp technique at the end of the fibre (Adrian, Chandler & Hodgkin, 1970a) has been employed. The two voltage-recording micro-electrodes V1 and V2 were placed at distances 1 and 21 respectively from the end of a superficial fibre. The currentinjecting micro-electrode was inserted at a distance 21 + 1' from the end of the fibre. In all experiments I = 0-47 mm and 1' = 0.12 mm. V1 was used for voltage control. Traces of V1, AV = V2-V1 and total current 1 were displayed on a storage oscilloscope and photographed. Linear electrical constants were calculated from cable theory using Ri = 169 Q cm (Hodgkin & Nakajima, 1972) for myoplasmic resistivity at 20 'C. Membrane currents were obtained from the approximate equation:
3Ri2 where a is the fibre radius and I is the interelectrode distance as defined before (Adrian et al. 1970a). Voltage steps were rounded with a time constant of 1 msec. Voltage-recording micro-electrodes were filled with 3 M-KCl; current-injecting micro-electrodes were filled with 2 M-K citrate. All micro-electrodes had resistances between 5 and 10 Mfl. Solutions contained sulphate as impermeant substitute for chloride. The composition of the bathing media was the following: Na2SO4 35 mm, CaSO4 8 mm, sucrose 113 mm, K2S04 0-5, 1, 2 or 5 mm, Na2HPO4 1-7 mm, NaH2PO4 1-1 mm, pH 6-9. With these solutions, in the range of potentials negative to the resting potential, K+ is essentially the only permeant ion. All the experiments were at room temperature, 20 + 2 'C.
RESULTS
Decline of conductance during hyperpolarization Fig. 1 shows the main features of the decline of conductance upon hyperpolarization. After dissection in Ringer solution the muscles were immersed in one of the solutions described in the Methods section at the desired K+ concentration. After equilibration for 1 hr in the new solution, voltage steps were applied to the fibres and the membrane currents recorded.
303 VOLTAGE GATING OF K+ INWARD RECTIFIER Fig. 1 refers to a fibre immersed in 4 mm external potassium. The decline of inward current is evident in Fig. 1 A. It is possible to see that although the initial current (immediately after the capacitative peak) increases with increasing hyperpolarizations, the steady-state current does not do so for large hyperpolarizations. A Al
V2
I_%
(--
v, VI-----
IF-
_,_~~ ~ ~ i
,/
8
Voltage (mV) -140
-120
E
0
-40 0
-60
Fig. 1. A, records showing the decline of inward current upon hyperpolarization. Calibration bars correspond to 200 msee, 100 mV and 56-4 ,sA/cm2. [K+]o 4 mM; resting and holding potential -80 mV. B, current-voltage relations from same fibre as A. Open circles (0) are currents immediately after capacitative peak at the onset of the voltage pulse (instantaneous currents). Filled circles (-) are steady-state currents. Note the negative slope conductance beyond - 140 mV.
This feature is shown better in Fig. 1 B where current-voltage curves are plotted for initial (instantaneous) current and steady-state current. The steady-state currentvoltage relation shows a negative slope for voltages more negative than - 140 mV.
E. MANCINELLI AND A. PERES
304
Effect of depletion on membrane conductance A decrease of potassium concentration in the tubular lumen will influence the membrane current in two ways: (a) as mentioned in the Introduction, the potassium equilibrium potential VK across the tubular wall will become more negative, decreasing the electrochemical gradient and (b) the membrane conductance will be smaller as an effect of the lower potassium concentration per se. Voltage (mV) -160 -6
-140
-100
-120
,0
-60
-80
/ ---y - w\
1
X-60
~~~~-20
D Vh -V
4
-40
C
Vc
41 ~~
~
~
~
~
~
-0
holding-60 p -80
/C
~~~~~~~0 A
-100
Fig. 2. Analysis of two-step experiments. The step sequence is shown on the left. At the end of a conditioning pulse, Ic, a test pulse to different potentials, V, was applied. Both pulses were 400 msec in duration. On the right, current-voltage relations from this experiment are shown. Curve A (0) is the instantaneous current (IA)-voltage relation of the conditioning pulse. Curve B (A) is the instantaneous current (It)-voltage relation after a conditioning pulse to -90 mV. Curve (lat) r is the same as B but after conditioning at - 120 mV. CurveD (V) is the same after conditioning at - 150 mV. Filled symbols are steady-state currents from the same steps. [K+]o 10 mm; resting and holding potentials -60 mV.
In order to gain some insight into these two effects we carried out a series of twostep experiments. As shown in the panel of Fig. 2 the membrane was first hyperpolarized by a conditioning step Vs; the step was 400 msec in duration in order to reach a steady current. At the end of this hyperpolarization test steps to different potentials and 400 msec in duration were applied. Several runs with different values of conditioning potential were performed on the same fibre. Results from one such experiment are shown in Fig. 2. Curve A shows the current~-voltage relation for instantaneous currents due to the conditioning steps (Ic)- Curves B, C and D show -current-voltage relations for instantaneous currents due to the test steps after conditioning steps to - 90, - 120 and - 150 mV respectively. The fibre was immersed in a 10 mm-K+ solution, which gave a resting potential of - 60 mV. The dashed curve represents the steady-state currents and it is evident that it is the same for conditioning and test steps; after a suitable time the membrane reaches a steady state which is independent of the previous history.
305 VOLTAGE GATING OF K+ INWARD RECTIFIER Two main features are apparent from this experiment: (a) the current-voltage relationships of the initial current after a conditioning pulse have a zero-current potential shifted toward more negative potentials, the shift being greater as the level of the conditioning potential is made more negative and (b) the slope of the linear portion of the relationships decreases with increasing conditioning hyperpolarizations. Voltage (mV)
-160
-140
-120
-100
=8
-
0
-20 E
-40 -
-60
0
-80
Fig. 3. Current-voltage relations obtained from Fig. 2 after subtraction from each curve of the 15 % of the instantaneous unconditioned current.
Both these observations are in agreement with theoretical expectations of the effects of a diminished K+ tubular concentration. The question now arises of whether the low K+ concentration in the tubules may explain quantitatively the experimental results. Almers (1972a, b) suggested that this is not the case. Studying the recovery of conductance after depleting pulses in a number of different experimental situations, he concluded that the conductance decline was far greater than would have been possible to predict from depletion alone. In order to have more direct evidence for this hypothesis we compared the decrease in conductance due to conditioning hyperpolarizations with the conductance of fibres immersed in various K+ concentrations. A test for the depletion hypothesis In this section we will make use of the hypothesis that K+ tubular depletion is the sole cause of the conductance decrement. Subsequently we shall compare the consequences of this hypothesis with the experimental observations. If there are no voltage-dependent permeability changes, for a given voltage step, the part of current flowing through the surface membrane may be taken as timeindependent, assuming that there are no changes of K+ concentration in the bulk
E. MANCINELLI AND A. PERES 306 solution. In this case current-voltage relations for tubular current only can be obtained. From the work of Almers (1972b) and also other authors (Eisenberg & Gage, 1969) it is estimated that the density of the inwardly rectifying channels for K+ ions is the same for surface and tubular membranes. This means that the greater part of the potassium current flows through the tubular membrane. Almers (1972b) estimated that at most 20 % of the total potassium current flows through the surface membrane. 1 _0a
0-8
00 0-6
0*4
-
-
00 0-2 _
0
0
I
-160
1
-140
I
1.
1
-120
1
1
1I -60 -80
-100 Conditioning potential (mV)
Fig. 4. Diagram illustrating the conductance decrease due to the conditioning step. Abscissa: value of the conditioning potential. Ordinate: ratio between conductance after conditioning pulse and normal conductance. Conductance values are obtained from the slope of the linear portion of the current-voltage relations as in Fig. 3. Re. sults from three fibres. [K+]o 10 mm.
We treated the results of Fig. 2 following this idea. Assuming that the surface current is at any potential 15 % of the initial current due to the conditioning step, we subtracted this part from all the instantaneous current-voltage curves of Fig. 2. Results of this treatment are shown in Fig. 3. These curves should now represent the relations between voltage and the tubular current only. It is evident that this procedure increases the effect on the shift of the zerocurrent potential and on the decrease of the slope conductance after conditioning
hyperpolarizations. The slope conductance measured in the linear portion of the current-voltage curves after conditioning steps relative to the value of the curve obtained without conditioning step is plotted in Fig. 4 against the conditioning potential. Data are from three fibres in 10 mm external potassium concentration. It is seen that maintaining the potential at hyperpolarized values for 400 msec
307 VOLTAGE GATING OF K+ INWARD RECTIFIER produces a progressive reduction of the tubular conductance which at - 150 mV reaches less than 20 % of the value at resting potential. In Fig. 5 is shown the shift of the zero-current potential Vz, as a function of the conditioning potential for the same three fibres of Fig. 4. According to the assumption made at the beginning of this paragraph, the zero-current potential may be taken as an indicator of the tubular potassium concentration. 50
0 40
a 0
_ 30
2-2
0
10
A
0 0
I
0
-160
-140
I3
-100 -120 Conditioning potential (mV)
00 I
I
-80
-60
Fig. 5. Diagram illustrating the shift of zero-current potential V., due to the conditioning step. Abscissa: value of the conditioning potential. Ordinate: absolute values of the shift obtained from graphs of the kind of Fig. 3. Same fibres as in Fig. 4.
Owing to the length and narrowness of the transverse tubular system the tubular potassium concentration will be non-uniform but it is probable (see p. 314) that the average depletion increases with increasing conditioning hyperpolarization (in the range we examined) and that in the steady state some current still flows through the tubular membrane, since the zero-current potential is always more positive than the value of the conditioning potential. The relative decrease in conductance and the shift in Vzc produced by the same conditioning pulse have been related to each other in Fig. 6 (open symbols). Data are those of Figs. 4 and 5. The membrane conductance in different external K+ concentrations The conductance of the fibre membrane has been measured as a function of the external potassium concentration in order to see whether the data in Fig. 6 could be explained simply by a reduction of the tubular potassium concentration. Experiments were done in the following way: after dissection muscles were immersed in a solution containing the desired concentration of K+ (1, 2 or 4 mM).
308 E. MANCINELLI AND A. PERES After equilibration for 1 hr resting potential, linear electrical constants and currentvoltage relations were measured in that solution for a number of fibres. Then the solution was changed to the one containing 10 mm-K+, which was used as a control, and other fibres were examined. 0
1
08 F
0-6 _
+ 0*4
_
B
0 0 2 _-
0
0
A I
0
-110
-120
i
-100
I
I
i
-90
-80
-70
-60
VZC (mV)
Fig. 6. Comparison between depletion effect and [K+]o effect. Open symbols are derived from Figs. 4 and 5. They show the relative conductance and the zero-current potential due to the same conditioning pulse. Filled symbols are values from Table 1, columns 3 and 8. Bars are S.E. of mean. TABLE 1. Effect of [K+]o on membrane conductance 1
2 (K+)o (mm)
3
4
mVI (mV)
T1
6 7 8 A Gm n (mV) (mm) (,umho/cm2) 11 10 -60+0-7 -60 28-4+2-6 1-04+0-05 1 779+53 7 4 -76+ 1-8 -76+ 1-7 27-4+ 1-3 0-67 1-30+0-06 519+49 2 5 - 90 - 90 32-0 + 4-9 1-68 + 0-15 327 + 28 0-42 9 1 -100+1-2 -101+0-6 26-8+1 1 1-46+0-04 375+21 0-48 n is the number of fibres; [K+]0 is the external potassium concentration; V< is the resting potential; T1 is the holding potential; a is the fibre radius; A is the fibre space constant; Gm is the membrane conductance per unit area; Go* is the membrane conductance per unit area relative to the value in 10 mM-[K+bO. Values are means + s.E. of the mean. 5 a ('I)
On some occasions micro-electrodes remained in a fibre during the solution change so that the same fibre was examined in the test and in the control solutions. Results are shown in Table 1. The holding potential was as always kept very close to the resting potential. The last column in Table 1 shows the changes in conductance relative to the value in the 10 mM-K+ solution.
309 VOLTAGE GATING OF K+ INWARD RECTIFIER It can be seen that the conductance value of the fibres immersed in 1 mM-K+ concentration is unexpectedly large. In the five experiments with this solution the fibres were easily damaged by the impalements. Although fibres which lost more than 10 mV of resting potential during impalement were methodically discarded, we think that this particular conductance value is mainly due to imperfect sealing of the membrane around the micro-electrodes. The lowest membrane conductance (Gm) value that we measured for a single fibre in 1 mM-K+ was 253 /amho/cm2 which corresponds to a value of 0-32 for G* (the membrane conductance relative to the value in 10 mM-K+). V1
0
C C0 dr
_
t
*%
ii
~~~~~d
Ring volume element
1i(r+dr)
dr
l
VI (r + dr)
-4-$ 1'(r)
V (r) {
Fig. 7. Schematic representation of the equivalent disk approximation of the transverse tubular system. In the upper part transverse and longitudinal sections of a disk showing the ring element used in the calculations. Lower part: detail of the current fluxes and of the potential changes in the ring element.
Comparison of two-step experiments with K+ concentration experiments With the results contained in Table 1 we are now in a position to verify the validity of the idea of the tubular potassium depletion as the sole cause of the conductance decrease. Data from Table 1, columns 3 and 8 are shown in Fig. 6 (filled symbols). For the reasons given above, curve B has been drawn using the lowest conductance value measured in 1 mM-K+, i.e. G* = 0-32 rather than the mean value of the nine fibres. Only if K+ concentration and potential were uniform in the tubules could we compare the two sets of results in Fig. 6. Indeed, at the end of a conditioning step, concentration and potential are likely to be very non-uniform there. The concen-
E. MANCINELLI AND A. PERES 310 tration non-uniformity together with the inward rectifier properties will tend to lower and to rotate toward the right curve B. In order to see whether the difference between curve B and curve A could be due to this effect we proceed to examine a model describing the situation of the tubules in the steady state.
K+ concentration and potential non-uniformities in the transverse tubule system Analytical model. This model is in many respects similar to the one developed by Barry & Adrian (1973).
The equivalent disk approximation of the transverse tubule system developed by Adrian et al. (1969) will be used: Fig. 7 shows two sections of a disk. We will further assume that in the steady state only potassium currents are present and that concentration changes in the sarcoplasm are negligible. During a voltage pulse potassium will move radially in the tubular lumen under the effect of concentration and potential gradients. The longitudinal current will then be given by: (1) i=- 2rrdFoD dr +RT C dr ' which is simply the Nernst-Planck equation for electrodiffusion and where r is the radial distance from the centre of the fibre, d is the thickness of the equivalent disk, oC is a dimensionless network factor introduced to account for the complexity of the transverse tubular system mesh, D is the potassium diffusion coefficient, C is the potassium concentration in the transverse tubular system lumen and VI is the potential difference between lumen and external solution; R, T and F have their usual meaning. The current crossing the tubular membrane through a ring element of thickness dr at a distance r from the centre will be: di = 4rrrdrrl, (2) where I' is the current density through the tubular membrane and is a function of potassium concentration and potential across the tubular membrane, i.e. II = Ij§, V, Vi), where V is the intracellular potential. Since d= (3) dr dr di
from eqns. (1), (2) and (3) and substituting d = p/n2nra (Barry & Adrian, 1973) where p is the fraction of fibre volume occupied by the tubules, a is the fibre radius and n is the number of disks per unit surface area (we take n = 1), we can write: d j JJ rK. 27Ta dr [dr rRT The potential changes in the radial direction will be given by Ohm's law: i dR, dV, =
(4)
(5)
VOLTAGE GATING OF K+ INWARD RECTIFIER 311 where dR is the electrical resistance of the ring element of thickness dr in the radial direction and it is given by: dr dR= (6) where GL is the conductance of the tubular lumen. Hence eqn. (5) becomes: dr
27Trdo-GL(
Substituting eqn. (1) in eqn. (7) we have:
d~l
FD dC F CdV1\ drCGLdr+RT dr and after some simplification: dl = FD/GL x d(8) dr -(1 -F2DC/RTGL) T'd(8
After integration and with the condition that Vi = 0 when C = centration in the external bulk solution, we get: = RTl I1-F2DC/RTGI LI
Co, i.e. the con-
VIRFIn 11-F2DCo/RTG
(9)
Substituting eqn. (8) in eqn. (4) and with some rearrangement: I dC\ pFo-D d / 2I 2lra dr (l -F2DC/RTG X7X-j-) =-2rI.K (10) Differentiating and simplifying: d2C F2D/RTGL (dC\2 1 dC _4na(1-F2DC/RTGL) It (11) dr2+ 1F2DC/RTGL kdr! +r dr paDF Eqn. (11) may be integrated if one has an analytical expression for IP as a function of potential and K+ concentration. Fig. 8 shows experimental curves of instantaneous potassium current against potential in four different external potassium concentrations, after subtraction of the 15 % in order to obtain the tubular current only. The procedure to construct an analytical expression for IP has been the following: the current-voltage curve in 1 mM-potassium has been fitted with a fifth order polynomial in the range between x
-170 and -60 mV, viz.: I([K+]o
=
1
a, V'5 + a2 V'4 + a3 V'8 + a4 V'2 + ar V'+ a6, For the fitting procedure VI = 0.
mM)
=
(12)
where V' = V- l-Vp,. Values of I for other potassium concentrations have been obtained from eqn. (12) shifting V., in agreement with the relation (Adrian, 1956): [K ]O+ a[Na+]O VIZC (13)
RTF-n
E. MANCINELLI AND A. PERES
312
10
Voltage (mV) -80 -140
-160
-120
-100
0
-20_ E
-40
'
-60
-80
Fig. 8. Current-voltage relationships obtained from the experiments with different external K+ concentrations. Each curve is the average of all curves obtained from the fibres listed in Table 1 for the corresponding concentration, except for [K+]o = 1 mm. For this concentration the curve is the individual curve of the fibre giving the lowest conductance value. All curves have been reduced by 15 % of their original value.
1
0-8 _
0-6 _
0-4 _
0-2
F I
-110
o -1 0o
I
-100
I
-90 VDC (mV)
I
-80
II
-60
-70
Fig. 9. Filled circles are the same as Fig. 6 except for the point at the extreme left which corresponds to the fibre giving the lowest conductance in [K+], = 1 mm. The 2621 x 10-5; G2 curve has been fitted to the points using eqn. (14) with G1 = -3-765 x 10-1; G4 = -4-232. -5-867 x 10o-; G3 -
=
=
VOLTAGE GATING OF K+ INWARD RECTIFIER 313 with x = 0-026, [Na+]o = 70 mm and [K+]i = 140 mm, and scaling eqn. (12) by the relative conductance factor G*. The factor 0* has been obtained as a function of V, fitting a cubic equation to the data of Table 1, columns 3 and 8: 2 + G* = GI V3 + (14) The experimental points and the curve described by eqn. (14) are shown in Fig. 9. I is then given by: I = G* x (aV'5+a2V'4+a3V'3+a4V'2+a.V'+a6) (15) -160
Voltage (mV) -120 -100
-140
-80
00 C0 = 1 mM
co mm
0 1-~ ~ ~ ~ ~ ~
-60
10
-~~~~~ ~
-20 E
4 mM
-40 7 mM
0
10 mM -J
-80
Fig. 10. Family of current-voltage curves described by eqn. (15). The values of the parameters of eqn. (12) used to fit the experimental curve for [K+]o = (0 = 1 mm in Fig. 8 are the following: a, = -2 597 x 10-10; a. = -1 810 x10-7; a3= -2902 x 10-5; a. = -1-661 x 10-3; ar = 1-823 x 10-1; a6 = -1-560x 10-2.
The family of curves described by eqn. (15) is shown in Fig. 10. The main objective of this procedure was to obtain a relation between relative conductance of the linear portion of the current-voltage curves and Vj,, similar to the experimental one, as it is shown in Fig. 9. Since we assume that there is only one transverse tubular system equivalent disk per unit surface area we have: 'K
27Ta2-
(16)
After substitution of eqns. (12), (13), (14), (15) and (16) in eqn. (11) it is possible to integrate it numerically. Since the boundary conditions (dC/drlr O= 0 and C(r = a) = C0) are not defined for the same value of r, the integration routine was coupled with an iterative procedure which changed the value of C at r = 0 until the value of C at r = a was approximated with a precision of one part in 105.
E. MANCINELLI AND A. PERES
314
Computation results Integration of eqn. (11) has been performed for different values of hyperpolarizing potential. Values of the parameters were the same as in Barry & Adrian (1973) except for the fibre radius which was a = 28 x 10-4 cm to meet our experimental conditions. They were: D = 1*6 x 10-5 cm2 sec-1; OL = 10-2 mho cm-'; p = 0 003; C = 0*5; RT/F = 0'025 V. Co was always 10 mm. Fig. 11 shows the steady-state concentration profiles due to different potential pulses. It is seen that depletion is relevant even for moderate hyperpolarizations. -
10
8E C
C C
0
E
4-
0 la.
0
5-6
11-2 16-8 Fibre radius (Mlm)
22-4
28
Fig. 11. Results of integration of eqn. (11). Shown are concentration profiles of potassium in the equivalent disk for three values of clamp potential in the steady state. The model did not permit the integration for potential values more negative than - 150 mV, since a negative concentration in r = 0 would have been required to obtain 0C = 10 mM.
The steady-state current for each potential pulse can be calculated from eqn. (1). Substituting in it eqn. (8) and after some rearrangements we have:
i(r) = -pFo1-Dal1-F2DC(r)/RTGL x dr
(17)
The steady-state current per unit surface area is equal to the longitudinal current calculated in r = a, hence: dC pFo-D (18) i(a) = - 1-F2DCO/RTL xdr r=a Fig. 12 shows curves of instantaneous current (obtained from eqn. (15)) and of steady-state current (obtained from eqn. (18)). It is clear that the current reduction
315
VOLTAGE GATING OF K+ INWARD RECTIFIER -160
-140
Voltage (mV) -120 -100
10 -80
-60
0
-20 E
-40
c
-60
-80
Fig. 12. Reconstruction of instantaneous and steady-state current-voltage relationships. The instantaneous current curve is obtained from eqn. (15) with C0 = 10 mM. The steady-state curve is obtained from eqn. (18) after integration of eqn. (11) had been performed. For the reasons given in the legend of Fig. 11 the steady-state curve goes only to -150 mV.
10
V,(mV) -160
-140
-120
-60
-100
-80
0
-20_ E V
=
-150 mV
-120 mV
-40 '
-90 mV
-60
-80 Fig. 13. Simulation of two-step experiments. This Figure is strictly analogous to Fig. 3. The three uppermost continuous curves represent the instantaneous current-voltage relationships for test steps after conditioning steps to three different potentials. The lowermost curve is the instantaneous current-voltage relationship without conditioning step. Dashed curve is the steady-state current. The computation has been done as explained in the text using eqn. (19).
E. MANCINELLI AND A. PERES
316 1
A
0
0-8 -
A
00 0-6 _
04
-
00 0-2 _
0o 0 nI | L- I
I
-160
-140
I
I
I
-100
-120
-80
I
-60
VI (mV) 50 B
0 40
-
0
30
E -
2
AS '20
0
0
A
0 10
0
A 0
-160
-140
-120
-100
-80
V, (mV)
Fig. 14. Comparison between experimental and computed results. Points are the same as in Figs. 4 and 5. A, the continuous line is the relative conductance decrease of the linear portion of the curves due to different conditioning potentials, obtained from Fig. 13. B, the continuous line is the shift of the zero-current potential due to different conditioning potentials, obtained from Fig. 13.
VOLTAGE GATING OF K+ IN WARD RECTIFIER 317 is much smaller than the experimental one (see Figs. 1 and 2) and also that there are no regions with negative slope conductance, as mentioned earlier. Two-step experiments simulation At the end of a conditioning step V = V,, when the clamp potential is moved abruptly to the test value Vt, because of the radial non-uniformities different patches of tubular membrane will be in situations described by different current-voltage curves in Fig. 10 and also at different voltage levels. Each patch will be crossed by a current given by eqn. (2). Integrating it from 0 to a and remembering eqn. (16) we obtain the total current flowing through the tubular membrane referred to the unit surface area:
I(Vc, Vt)
=-2
rI(r) dr.
(19)
This is the instantaneous current due to a test step to Vt after a conditioning step to Vc. A computer programme has been written which, while calculating the concentration profile for a given V = Ve, estimated the right-hand side of eqn. (19) for different values of Vt between - 170 and -60 mV. The function I(V,, Vt) is shown in Fig. 13. These curves are now comparable with the ones shown in Fig. 3. It is seen that the effect caused by the conditioning step is smaller both on the zero-current potential shift and on the conductance decrease. This is better shown in Fig. 14 where experimental and computed results are plotted together. DISCUSSION
Although no complete reliance can be given to the model, the difference between experimental and computed results in Fig. 14 is so significant that the values of the parameters or the theoretical assumptions or both would have to be changed drastically in order to overcome this discrepancy. The difference in the results in Fig. 14 can be mainly imputed to the assumption that there are no voltage-dependent permeability changes. If the presence of a voltage-dependent gate in series with the potassium channel is postulated both for surface and tubular membrane, then Fig. 3 will not represent correctly the current flowing through the tubular membrane. The surface current at any potential would depend on the degree of inactivation of the gate at the conditioning potential. From a qualitative point of view, to obtain the tubular current alone we should subtract less than 15 % of the instantaneous current due to the conditioning steps. This correction will reduce the shift of the zero-current potential and also the amount of relative conductance decrease in the experimental points in Fig. 14. The presence of an access resistance at the mouth of the tubules (Adrian & Peachey, 1973) has not been considered in our model. This resistance would tend to reduce non-uniformities in the tubular lumen. In such a case the relation between relative conductance and zero-current potential that would be theoretically expected would lie very close to curve B in Fig. 6, increasing the disagreement between experimental and computed results. The disagreement depends also on the value of the part of the total current which
E. MIANCINELLI AND A. PERES is estimated to flow through the surface membrane at the beginning of the conditioning steps. The higher this value the greater the disagreement. The value of 15 % that we used is already a lower limit, compared to the estimates of Almers (1972b) and of Eisenberg & Gage (1969). The results of the present paper confirm the observation made by Almers (1972b) that the conductance changes after hyperpolarization cannot be wholly due to depletion, the further reduction being imputable to a voltage-dependent permeability change. Studying the recovery of conductance after a depleting pulse, Almers was able to construct the steady-state activation curve for this gate. His Fig. 5 shows the shape of this curve (Pa,) which he found to be equal to 1 (fully open gate) at resting potential and to begin to decrease with hyperpolarizations beyond 120 mV. While our experiments confirm the existence of a voltage-dependent gate, we are unable to estimate the shape of its activation curve. In fact only if the gate were not present in the surface membrane would we be able to obtain POL as the ratio between the experimental and the computed points in Fig. 14A. It is apparent in Fig. 14 that the difference between experimental and computed results is already significant at values of conditioning potential more positive than - 120 mV. Though this could be due to the model, our results suggest that the position of the PO,, curve on the voltage axis might be shifted a little to the right with respect to the curve given by Almers (1972b). 318
-
The authors would like to thank Dr R. H. Adrian for helpful criticism, Professor A. Ferroni and Dr F. Andrietti for reading the manuscript and the Centro Ricerca di Automatica ENEL for assistance in the computations. This work has been partially supported by a CNR research grant, No. 78.02099.04. REFERENCES
ADRIAN, R. H. (1956). The effect of internal and external potassium concentration on the membrane potential of frog muscle. J. Physiol. 133, 631-658. ADRIAN, R. H., CHANDLER, W. K. & HODGKIN, A. L. (1969). The kinetics of mechanical activation in frog muscle. J. Physiol. 204, 207-230. ADRIAN, R. H., CHANDLER, W. K. & HODGKIN, A. L. (1970a). Voltage clamp experiments in striated muscle fibres. J. Physiol. 208, 607-644. ADRIAN, R. H., CHANDLER, W. K. & HODGKIN, A. L. (1970b). Slow changes in potassium permeability in skeletal muscle. J. Physiol. 208, 645-668. ADRIAN, R. H. & FREYGANG, W. H. (1962). The potassium and chloride conductances of frog muscle. J. Physiol. 163, 61-103. ADRIAN, R. H. & PEACHEY, L. D. (1973). Reconstruction of the action potential of frog sartorius muscle. J. Physiol. 235, 103-131. ALMERS, W. (1972a). Potassium conductance changes in skeletal muscle and the potassium concentration in the transverse tubules. J. Physiol. 225, 33-56. ALMERS, WV. (1972b). The decline of potassium permeability during extreme hyperpolarization in frog skeletal muscle. J. Physiol. 225, 57-83. BARRY, P. H. & ADRIAN, R. H. (1973). Slow conductance changes due to potassium depletion in the transverse tubules of frog muscle fibres during hyperpolarizing pulses. J. Membrane Biol. 14, 243-292. EISENBERG, R. S. & GAGE, P. W. (1969). Ionic conductances of the surface and transverse tubular membrane of frog sartorius fibres. J. gen. Physiol. 53, 279-297. HODGKIN, A. L. & HOROWICZ, P. (1960). The effect of sudden changes in ionic concentrations on the membrane potential of single muscle fibres. J. Physiol. 153, 370-385. HODGKIN, A. L. & NAKAJIMA,S. (1972). The effect of diameter on the electrical constants of frog skeletal muscle fibres. J. Physiol. 221, 105-120. NAKAJIMA, S., NAKAJIMA, Y. & PEACIEY, L. D. (1969). Speed of repolarization and morphology of glycerol-treated muscle fibres. J. Phy.siol. 200, 115-1 16P.
301
With 14 text-ftgures Printed in Great Britain
A VOLTAGE-DEPENDENT GATE IN SERIES WITH THE INWARDLY RECTIFYING POTASSIUM CHANNEL IN FROG STRIATED MUSCLE
BY E. MANCINELLI AND A. PERES From the Istituto di Fisiologia Generale e Chimica Biologica, Universitd di Milano, Via Mangiagalli, 32, Milano, Italy
(Received 25 April 1978) SUMMARY
1. The degree of tubular potassium depletion and the decrease of potassium conductance due to hyperpolarizing pulses in striated muscular fibres have been examined with the three micro-electrode voltage-clamp technique. 2. The conductance of the fibre membrane has been measured in different extracellular K+ concentrations from 1 to 10 mm. 3. Comparison of the two sets of measurements shows that changes in tubular K+ concentration are not sufficient to account for the conductance decrease associated with hyperpolarization. 4. The presence of a voltage-dependent gate in series with the inwardly rectifying channel for K+ ions, suggested by Almers (1972a, b), is thus confirmed. INTRODUCTION
In recent years K+ depletion or accumulation in extracellular spaces has received considerable attention. When the membrane of a striated muscle fibre is hyperpolarized, the inward current declines to reach a steady level (Adrian & Freygang, 1962; Adrian, Chandler & Hodgkin, 1970b). In this preparation, in the range of potentials negative to the resting potential, only the inwardly rectifying channel for K+ seems to be operative. While for hyperpolarizing pulses the instantaneous current voltage relation is reasonably linear, the steady-state relation shows a region of negative slope conductance- for extreme hyperpolarizations. This feature has been already discussed by Adrian et al. (1970b) and by Almers (1972a, b). These authors pointed out that it is difficult to explain the negative conductance simply on the basis of a decrease of potassium concentration in the transverse tubules. Indeed, since the inward rectifier resides mostly in the tubular membrane (Nakajima, Nakajima & Peachey, 1969; Hodgkin & Horowicz, 1960), the passage of inward current would cause a decrease in the tubular K+ concentration (depletion) if the replenishment of the tubules from the bulk solution is limited by a low transport number for potassium. This would make the K+ equilibrium potential across the tubular membrane more negative, leading to a decrease of the driving force V - VK. The tubular potassium current then should decrease with increasing depletion. 0022-3751/79/4200-0423 $01.50 © 1979 The Physiological Society
302 E. MANCINELLI AND A. PERES In the steady state, however, the K+ current leaving the tubules through the membrane must equal the current entering the tubules from the bulk solution. If one makes the reasonable assumption that the current entering is proportional to the concentration difference between bulk solution and the tubular lumen, the steady-state current should increase with increasing depletion. To explain this inconsistency with his experimental results, Almers (1972a, b) suggested, in addition to the tubular depletion, the presence of a voltage-dependent gate in series with the inwardly rectifying channel. This structure, while fully open at the resting potential, tends to close when the potential is made more negative than - 120 mV. In the experiments reported here the existence of this voltage-dependent gate has been examined. We compared the conductance of fibres previously depleted by hyperpolarizing pulses with the conductance of non-depleted fibres immersed in 'different external K+ concentrations. The results show that changes in tubular K+ concentration are insufficient to account for the conductance decline and favour the hypothesis of the existence of a voltage-dependent gate. METHODS Whole semitendinosus muscles of Rana ewculenta were dissected out. The three micro-electrode voltage-clamp technique at the end of the fibre (Adrian, Chandler & Hodgkin, 1970a) has been employed. The two voltage-recording micro-electrodes V1 and V2 were placed at distances 1 and 21 respectively from the end of a superficial fibre. The currentinjecting micro-electrode was inserted at a distance 21 + 1' from the end of the fibre. In all experiments I = 0-47 mm and 1' = 0.12 mm. V1 was used for voltage control. Traces of V1, AV = V2-V1 and total current 1 were displayed on a storage oscilloscope and photographed. Linear electrical constants were calculated from cable theory using Ri = 169 Q cm (Hodgkin & Nakajima, 1972) for myoplasmic resistivity at 20 'C. Membrane currents were obtained from the approximate equation:
3Ri2 where a is the fibre radius and I is the interelectrode distance as defined before (Adrian et al. 1970a). Voltage steps were rounded with a time constant of 1 msec. Voltage-recording micro-electrodes were filled with 3 M-KCl; current-injecting micro-electrodes were filled with 2 M-K citrate. All micro-electrodes had resistances between 5 and 10 Mfl. Solutions contained sulphate as impermeant substitute for chloride. The composition of the bathing media was the following: Na2SO4 35 mm, CaSO4 8 mm, sucrose 113 mm, K2S04 0-5, 1, 2 or 5 mm, Na2HPO4 1-7 mm, NaH2PO4 1-1 mm, pH 6-9. With these solutions, in the range of potentials negative to the resting potential, K+ is essentially the only permeant ion. All the experiments were at room temperature, 20 + 2 'C.
RESULTS
Decline of conductance during hyperpolarization Fig. 1 shows the main features of the decline of conductance upon hyperpolarization. After dissection in Ringer solution the muscles were immersed in one of the solutions described in the Methods section at the desired K+ concentration. After equilibration for 1 hr in the new solution, voltage steps were applied to the fibres and the membrane currents recorded.
303 VOLTAGE GATING OF K+ INWARD RECTIFIER Fig. 1 refers to a fibre immersed in 4 mm external potassium. The decline of inward current is evident in Fig. 1 A. It is possible to see that although the initial current (immediately after the capacitative peak) increases with increasing hyperpolarizations, the steady-state current does not do so for large hyperpolarizations. A Al
V2
I_%
(--
v, VI-----
IF-
_,_~~ ~ ~ i
,/
8
Voltage (mV) -140
-120
E
0
-40 0
-60
Fig. 1. A, records showing the decline of inward current upon hyperpolarization. Calibration bars correspond to 200 msee, 100 mV and 56-4 ,sA/cm2. [K+]o 4 mM; resting and holding potential -80 mV. B, current-voltage relations from same fibre as A. Open circles (0) are currents immediately after capacitative peak at the onset of the voltage pulse (instantaneous currents). Filled circles (-) are steady-state currents. Note the negative slope conductance beyond - 140 mV.
This feature is shown better in Fig. 1 B where current-voltage curves are plotted for initial (instantaneous) current and steady-state current. The steady-state currentvoltage relation shows a negative slope for voltages more negative than - 140 mV.
E. MANCINELLI AND A. PERES
304
Effect of depletion on membrane conductance A decrease of potassium concentration in the tubular lumen will influence the membrane current in two ways: (a) as mentioned in the Introduction, the potassium equilibrium potential VK across the tubular wall will become more negative, decreasing the electrochemical gradient and (b) the membrane conductance will be smaller as an effect of the lower potassium concentration per se. Voltage (mV) -160 -6
-140
-100
-120
,0
-60
-80
/ ---y - w\
1
X-60
~~~~-20
D Vh -V
4
-40
C
Vc
41 ~~
~
~
~
~
~
-0
holding-60 p -80
/C
~~~~~~~0 A
-100
Fig. 2. Analysis of two-step experiments. The step sequence is shown on the left. At the end of a conditioning pulse, Ic, a test pulse to different potentials, V, was applied. Both pulses were 400 msec in duration. On the right, current-voltage relations from this experiment are shown. Curve A (0) is the instantaneous current (IA)-voltage relation of the conditioning pulse. Curve B (A) is the instantaneous current (It)-voltage relation after a conditioning pulse to -90 mV. Curve (lat) r is the same as B but after conditioning at - 120 mV. CurveD (V) is the same after conditioning at - 150 mV. Filled symbols are steady-state currents from the same steps. [K+]o 10 mm; resting and holding potentials -60 mV.
In order to gain some insight into these two effects we carried out a series of twostep experiments. As shown in the panel of Fig. 2 the membrane was first hyperpolarized by a conditioning step Vs; the step was 400 msec in duration in order to reach a steady current. At the end of this hyperpolarization test steps to different potentials and 400 msec in duration were applied. Several runs with different values of conditioning potential were performed on the same fibre. Results from one such experiment are shown in Fig. 2. Curve A shows the current~-voltage relation for instantaneous currents due to the conditioning steps (Ic)- Curves B, C and D show -current-voltage relations for instantaneous currents due to the test steps after conditioning steps to - 90, - 120 and - 150 mV respectively. The fibre was immersed in a 10 mm-K+ solution, which gave a resting potential of - 60 mV. The dashed curve represents the steady-state currents and it is evident that it is the same for conditioning and test steps; after a suitable time the membrane reaches a steady state which is independent of the previous history.
305 VOLTAGE GATING OF K+ INWARD RECTIFIER Two main features are apparent from this experiment: (a) the current-voltage relationships of the initial current after a conditioning pulse have a zero-current potential shifted toward more negative potentials, the shift being greater as the level of the conditioning potential is made more negative and (b) the slope of the linear portion of the relationships decreases with increasing conditioning hyperpolarizations. Voltage (mV)
-160
-140
-120
-100
=8
-
0
-20 E
-40 -
-60
0
-80
Fig. 3. Current-voltage relations obtained from Fig. 2 after subtraction from each curve of the 15 % of the instantaneous unconditioned current.
Both these observations are in agreement with theoretical expectations of the effects of a diminished K+ tubular concentration. The question now arises of whether the low K+ concentration in the tubules may explain quantitatively the experimental results. Almers (1972a, b) suggested that this is not the case. Studying the recovery of conductance after depleting pulses in a number of different experimental situations, he concluded that the conductance decline was far greater than would have been possible to predict from depletion alone. In order to have more direct evidence for this hypothesis we compared the decrease in conductance due to conditioning hyperpolarizations with the conductance of fibres immersed in various K+ concentrations. A test for the depletion hypothesis In this section we will make use of the hypothesis that K+ tubular depletion is the sole cause of the conductance decrement. Subsequently we shall compare the consequences of this hypothesis with the experimental observations. If there are no voltage-dependent permeability changes, for a given voltage step, the part of current flowing through the surface membrane may be taken as timeindependent, assuming that there are no changes of K+ concentration in the bulk
E. MANCINELLI AND A. PERES 306 solution. In this case current-voltage relations for tubular current only can be obtained. From the work of Almers (1972b) and also other authors (Eisenberg & Gage, 1969) it is estimated that the density of the inwardly rectifying channels for K+ ions is the same for surface and tubular membranes. This means that the greater part of the potassium current flows through the tubular membrane. Almers (1972b) estimated that at most 20 % of the total potassium current flows through the surface membrane. 1 _0a
0-8
00 0-6
0*4
-
-
00 0-2 _
0
0
I
-160
1
-140
I
1.
1
-120
1
1
1I -60 -80
-100 Conditioning potential (mV)
Fig. 4. Diagram illustrating the conductance decrease due to the conditioning step. Abscissa: value of the conditioning potential. Ordinate: ratio between conductance after conditioning pulse and normal conductance. Conductance values are obtained from the slope of the linear portion of the current-voltage relations as in Fig. 3. Re. sults from three fibres. [K+]o 10 mm.
We treated the results of Fig. 2 following this idea. Assuming that the surface current is at any potential 15 % of the initial current due to the conditioning step, we subtracted this part from all the instantaneous current-voltage curves of Fig. 2. Results of this treatment are shown in Fig. 3. These curves should now represent the relations between voltage and the tubular current only. It is evident that this procedure increases the effect on the shift of the zerocurrent potential and on the decrease of the slope conductance after conditioning
hyperpolarizations. The slope conductance measured in the linear portion of the current-voltage curves after conditioning steps relative to the value of the curve obtained without conditioning step is plotted in Fig. 4 against the conditioning potential. Data are from three fibres in 10 mm external potassium concentration. It is seen that maintaining the potential at hyperpolarized values for 400 msec
307 VOLTAGE GATING OF K+ INWARD RECTIFIER produces a progressive reduction of the tubular conductance which at - 150 mV reaches less than 20 % of the value at resting potential. In Fig. 5 is shown the shift of the zero-current potential Vz, as a function of the conditioning potential for the same three fibres of Fig. 4. According to the assumption made at the beginning of this paragraph, the zero-current potential may be taken as an indicator of the tubular potassium concentration. 50
0 40
a 0
_ 30
2-2
0
10
A
0 0
I
0
-160
-140
I3
-100 -120 Conditioning potential (mV)
00 I
I
-80
-60
Fig. 5. Diagram illustrating the shift of zero-current potential V., due to the conditioning step. Abscissa: value of the conditioning potential. Ordinate: absolute values of the shift obtained from graphs of the kind of Fig. 3. Same fibres as in Fig. 4.
Owing to the length and narrowness of the transverse tubular system the tubular potassium concentration will be non-uniform but it is probable (see p. 314) that the average depletion increases with increasing conditioning hyperpolarization (in the range we examined) and that in the steady state some current still flows through the tubular membrane, since the zero-current potential is always more positive than the value of the conditioning potential. The relative decrease in conductance and the shift in Vzc produced by the same conditioning pulse have been related to each other in Fig. 6 (open symbols). Data are those of Figs. 4 and 5. The membrane conductance in different external K+ concentrations The conductance of the fibre membrane has been measured as a function of the external potassium concentration in order to see whether the data in Fig. 6 could be explained simply by a reduction of the tubular potassium concentration. Experiments were done in the following way: after dissection muscles were immersed in a solution containing the desired concentration of K+ (1, 2 or 4 mM).
308 E. MANCINELLI AND A. PERES After equilibration for 1 hr resting potential, linear electrical constants and currentvoltage relations were measured in that solution for a number of fibres. Then the solution was changed to the one containing 10 mm-K+, which was used as a control, and other fibres were examined. 0
1
08 F
0-6 _
+ 0*4
_
B
0 0 2 _-
0
0
A I
0
-110
-120
i
-100
I
I
i
-90
-80
-70
-60
VZC (mV)
Fig. 6. Comparison between depletion effect and [K+]o effect. Open symbols are derived from Figs. 4 and 5. They show the relative conductance and the zero-current potential due to the same conditioning pulse. Filled symbols are values from Table 1, columns 3 and 8. Bars are S.E. of mean. TABLE 1. Effect of [K+]o on membrane conductance 1
2 (K+)o (mm)
3
4
mVI (mV)
T1
6 7 8 A Gm n (mV) (mm) (,umho/cm2) 11 10 -60+0-7 -60 28-4+2-6 1-04+0-05 1 779+53 7 4 -76+ 1-8 -76+ 1-7 27-4+ 1-3 0-67 1-30+0-06 519+49 2 5 - 90 - 90 32-0 + 4-9 1-68 + 0-15 327 + 28 0-42 9 1 -100+1-2 -101+0-6 26-8+1 1 1-46+0-04 375+21 0-48 n is the number of fibres; [K+]0 is the external potassium concentration; V< is the resting potential; T1 is the holding potential; a is the fibre radius; A is the fibre space constant; Gm is the membrane conductance per unit area; Go* is the membrane conductance per unit area relative to the value in 10 mM-[K+bO. Values are means + s.E. of the mean. 5 a ('I)
On some occasions micro-electrodes remained in a fibre during the solution change so that the same fibre was examined in the test and in the control solutions. Results are shown in Table 1. The holding potential was as always kept very close to the resting potential. The last column in Table 1 shows the changes in conductance relative to the value in the 10 mM-K+ solution.
309 VOLTAGE GATING OF K+ INWARD RECTIFIER It can be seen that the conductance value of the fibres immersed in 1 mM-K+ concentration is unexpectedly large. In the five experiments with this solution the fibres were easily damaged by the impalements. Although fibres which lost more than 10 mV of resting potential during impalement were methodically discarded, we think that this particular conductance value is mainly due to imperfect sealing of the membrane around the micro-electrodes. The lowest membrane conductance (Gm) value that we measured for a single fibre in 1 mM-K+ was 253 /amho/cm2 which corresponds to a value of 0-32 for G* (the membrane conductance relative to the value in 10 mM-K+). V1
0
C C0 dr
_
t
*%
ii
~~~~~d
Ring volume element
1i(r+dr)
dr
l
VI (r + dr)
-4-$ 1'(r)
V (r) {
Fig. 7. Schematic representation of the equivalent disk approximation of the transverse tubular system. In the upper part transverse and longitudinal sections of a disk showing the ring element used in the calculations. Lower part: detail of the current fluxes and of the potential changes in the ring element.
Comparison of two-step experiments with K+ concentration experiments With the results contained in Table 1 we are now in a position to verify the validity of the idea of the tubular potassium depletion as the sole cause of the conductance decrease. Data from Table 1, columns 3 and 8 are shown in Fig. 6 (filled symbols). For the reasons given above, curve B has been drawn using the lowest conductance value measured in 1 mM-K+, i.e. G* = 0-32 rather than the mean value of the nine fibres. Only if K+ concentration and potential were uniform in the tubules could we compare the two sets of results in Fig. 6. Indeed, at the end of a conditioning step, concentration and potential are likely to be very non-uniform there. The concen-
E. MANCINELLI AND A. PERES 310 tration non-uniformity together with the inward rectifier properties will tend to lower and to rotate toward the right curve B. In order to see whether the difference between curve B and curve A could be due to this effect we proceed to examine a model describing the situation of the tubules in the steady state.
K+ concentration and potential non-uniformities in the transverse tubule system Analytical model. This model is in many respects similar to the one developed by Barry & Adrian (1973).
The equivalent disk approximation of the transverse tubule system developed by Adrian et al. (1969) will be used: Fig. 7 shows two sections of a disk. We will further assume that in the steady state only potassium currents are present and that concentration changes in the sarcoplasm are negligible. During a voltage pulse potassium will move radially in the tubular lumen under the effect of concentration and potential gradients. The longitudinal current will then be given by: (1) i=- 2rrdFoD dr +RT C dr ' which is simply the Nernst-Planck equation for electrodiffusion and where r is the radial distance from the centre of the fibre, d is the thickness of the equivalent disk, oC is a dimensionless network factor introduced to account for the complexity of the transverse tubular system mesh, D is the potassium diffusion coefficient, C is the potassium concentration in the transverse tubular system lumen and VI is the potential difference between lumen and external solution; R, T and F have their usual meaning. The current crossing the tubular membrane through a ring element of thickness dr at a distance r from the centre will be: di = 4rrrdrrl, (2) where I' is the current density through the tubular membrane and is a function of potassium concentration and potential across the tubular membrane, i.e. II = Ij§, V, Vi), where V is the intracellular potential. Since d= (3) dr dr di
from eqns. (1), (2) and (3) and substituting d = p/n2nra (Barry & Adrian, 1973) where p is the fraction of fibre volume occupied by the tubules, a is the fibre radius and n is the number of disks per unit surface area (we take n = 1), we can write: d j JJ rK. 27Ta dr [dr rRT The potential changes in the radial direction will be given by Ohm's law: i dR, dV, =
(4)
(5)
VOLTAGE GATING OF K+ INWARD RECTIFIER 311 where dR is the electrical resistance of the ring element of thickness dr in the radial direction and it is given by: dr dR= (6) where GL is the conductance of the tubular lumen. Hence eqn. (5) becomes: dr
27Trdo-GL(
Substituting eqn. (1) in eqn. (7) we have:
d~l
FD dC F CdV1\ drCGLdr+RT dr and after some simplification: dl = FD/GL x d(8) dr -(1 -F2DC/RTGL) T'd(8
After integration and with the condition that Vi = 0 when C = centration in the external bulk solution, we get: = RTl I1-F2DC/RTGI LI
Co, i.e. the con-
VIRFIn 11-F2DCo/RTG
(9)
Substituting eqn. (8) in eqn. (4) and with some rearrangement: I dC\ pFo-D d / 2I 2lra dr (l -F2DC/RTG X7X-j-) =-2rI.K (10) Differentiating and simplifying: d2C F2D/RTGL (dC\2 1 dC _4na(1-F2DC/RTGL) It (11) dr2+ 1F2DC/RTGL kdr! +r dr paDF Eqn. (11) may be integrated if one has an analytical expression for IP as a function of potential and K+ concentration. Fig. 8 shows experimental curves of instantaneous potassium current against potential in four different external potassium concentrations, after subtraction of the 15 % in order to obtain the tubular current only. The procedure to construct an analytical expression for IP has been the following: the current-voltage curve in 1 mM-potassium has been fitted with a fifth order polynomial in the range between x
-170 and -60 mV, viz.: I([K+]o
=
1
a, V'5 + a2 V'4 + a3 V'8 + a4 V'2 + ar V'+ a6, For the fitting procedure VI = 0.
mM)
=
(12)
where V' = V- l-Vp,. Values of I for other potassium concentrations have been obtained from eqn. (12) shifting V., in agreement with the relation (Adrian, 1956): [K ]O+ a[Na+]O VIZC (13)
RTF-n
E. MANCINELLI AND A. PERES
312
10
Voltage (mV) -80 -140
-160
-120
-100
0
-20_ E
-40
'
-60
-80
Fig. 8. Current-voltage relationships obtained from the experiments with different external K+ concentrations. Each curve is the average of all curves obtained from the fibres listed in Table 1 for the corresponding concentration, except for [K+]o = 1 mm. For this concentration the curve is the individual curve of the fibre giving the lowest conductance value. All curves have been reduced by 15 % of their original value.
1
0-8 _
0-6 _
0-4 _
0-2
F I
-110
o -1 0o
I
-100
I
-90 VDC (mV)
I
-80
II
-60
-70
Fig. 9. Filled circles are the same as Fig. 6 except for the point at the extreme left which corresponds to the fibre giving the lowest conductance in [K+], = 1 mm. The 2621 x 10-5; G2 curve has been fitted to the points using eqn. (14) with G1 = -3-765 x 10-1; G4 = -4-232. -5-867 x 10o-; G3 -
=
=
VOLTAGE GATING OF K+ INWARD RECTIFIER 313 with x = 0-026, [Na+]o = 70 mm and [K+]i = 140 mm, and scaling eqn. (12) by the relative conductance factor G*. The factor 0* has been obtained as a function of V, fitting a cubic equation to the data of Table 1, columns 3 and 8: 2 + G* = GI V3 + (14) The experimental points and the curve described by eqn. (14) are shown in Fig. 9. I is then given by: I = G* x (aV'5+a2V'4+a3V'3+a4V'2+a.V'+a6) (15) -160
Voltage (mV) -120 -100
-140
-80
00 C0 = 1 mM
co mm
0 1-~ ~ ~ ~ ~ ~
-60
10
-~~~~~ ~
-20 E
4 mM
-40 7 mM
0
10 mM -J
-80
Fig. 10. Family of current-voltage curves described by eqn. (15). The values of the parameters of eqn. (12) used to fit the experimental curve for [K+]o = (0 = 1 mm in Fig. 8 are the following: a, = -2 597 x 10-10; a. = -1 810 x10-7; a3= -2902 x 10-5; a. = -1-661 x 10-3; ar = 1-823 x 10-1; a6 = -1-560x 10-2.
The family of curves described by eqn. (15) is shown in Fig. 10. The main objective of this procedure was to obtain a relation between relative conductance of the linear portion of the current-voltage curves and Vj,, similar to the experimental one, as it is shown in Fig. 9. Since we assume that there is only one transverse tubular system equivalent disk per unit surface area we have: 'K
27Ta2-
(16)
After substitution of eqns. (12), (13), (14), (15) and (16) in eqn. (11) it is possible to integrate it numerically. Since the boundary conditions (dC/drlr O= 0 and C(r = a) = C0) are not defined for the same value of r, the integration routine was coupled with an iterative procedure which changed the value of C at r = 0 until the value of C at r = a was approximated with a precision of one part in 105.
E. MANCINELLI AND A. PERES
314
Computation results Integration of eqn. (11) has been performed for different values of hyperpolarizing potential. Values of the parameters were the same as in Barry & Adrian (1973) except for the fibre radius which was a = 28 x 10-4 cm to meet our experimental conditions. They were: D = 1*6 x 10-5 cm2 sec-1; OL = 10-2 mho cm-'; p = 0 003; C = 0*5; RT/F = 0'025 V. Co was always 10 mm. Fig. 11 shows the steady-state concentration profiles due to different potential pulses. It is seen that depletion is relevant even for moderate hyperpolarizations. -
10
8E C
C C
0
E
4-
0 la.
0
5-6
11-2 16-8 Fibre radius (Mlm)
22-4
28
Fig. 11. Results of integration of eqn. (11). Shown are concentration profiles of potassium in the equivalent disk for three values of clamp potential in the steady state. The model did not permit the integration for potential values more negative than - 150 mV, since a negative concentration in r = 0 would have been required to obtain 0C = 10 mM.
The steady-state current for each potential pulse can be calculated from eqn. (1). Substituting in it eqn. (8) and after some rearrangements we have:
i(r) = -pFo1-Dal1-F2DC(r)/RTGL x dr
(17)
The steady-state current per unit surface area is equal to the longitudinal current calculated in r = a, hence: dC pFo-D (18) i(a) = - 1-F2DCO/RTL xdr r=a Fig. 12 shows curves of instantaneous current (obtained from eqn. (15)) and of steady-state current (obtained from eqn. (18)). It is clear that the current reduction
315
VOLTAGE GATING OF K+ INWARD RECTIFIER -160
-140
Voltage (mV) -120 -100
10 -80
-60
0
-20 E
-40
c
-60
-80
Fig. 12. Reconstruction of instantaneous and steady-state current-voltage relationships. The instantaneous current curve is obtained from eqn. (15) with C0 = 10 mM. The steady-state curve is obtained from eqn. (18) after integration of eqn. (11) had been performed. For the reasons given in the legend of Fig. 11 the steady-state curve goes only to -150 mV.
10
V,(mV) -160
-140
-120
-60
-100
-80
0
-20_ E V
=
-150 mV
-120 mV
-40 '
-90 mV
-60
-80 Fig. 13. Simulation of two-step experiments. This Figure is strictly analogous to Fig. 3. The three uppermost continuous curves represent the instantaneous current-voltage relationships for test steps after conditioning steps to three different potentials. The lowermost curve is the instantaneous current-voltage relationship without conditioning step. Dashed curve is the steady-state current. The computation has been done as explained in the text using eqn. (19).
E. MANCINELLI AND A. PERES
316 1
A
0
0-8 -
A
00 0-6 _
04
-
00 0-2 _
0o 0 nI | L- I
I
-160
-140
I
I
I
-100
-120
-80
I
-60
VI (mV) 50 B
0 40
-
0
30
E -
2
AS '20
0
0
A
0 10
0
A 0
-160
-140
-120
-100
-80
V, (mV)
Fig. 14. Comparison between experimental and computed results. Points are the same as in Figs. 4 and 5. A, the continuous line is the relative conductance decrease of the linear portion of the curves due to different conditioning potentials, obtained from Fig. 13. B, the continuous line is the shift of the zero-current potential due to different conditioning potentials, obtained from Fig. 13.
VOLTAGE GATING OF K+ IN WARD RECTIFIER 317 is much smaller than the experimental one (see Figs. 1 and 2) and also that there are no regions with negative slope conductance, as mentioned earlier. Two-step experiments simulation At the end of a conditioning step V = V,, when the clamp potential is moved abruptly to the test value Vt, because of the radial non-uniformities different patches of tubular membrane will be in situations described by different current-voltage curves in Fig. 10 and also at different voltage levels. Each patch will be crossed by a current given by eqn. (2). Integrating it from 0 to a and remembering eqn. (16) we obtain the total current flowing through the tubular membrane referred to the unit surface area:
I(Vc, Vt)
=-2
rI(r) dr.
(19)
This is the instantaneous current due to a test step to Vt after a conditioning step to Vc. A computer programme has been written which, while calculating the concentration profile for a given V = Ve, estimated the right-hand side of eqn. (19) for different values of Vt between - 170 and -60 mV. The function I(V,, Vt) is shown in Fig. 13. These curves are now comparable with the ones shown in Fig. 3. It is seen that the effect caused by the conditioning step is smaller both on the zero-current potential shift and on the conductance decrease. This is better shown in Fig. 14 where experimental and computed results are plotted together. DISCUSSION
Although no complete reliance can be given to the model, the difference between experimental and computed results in Fig. 14 is so significant that the values of the parameters or the theoretical assumptions or both would have to be changed drastically in order to overcome this discrepancy. The difference in the results in Fig. 14 can be mainly imputed to the assumption that there are no voltage-dependent permeability changes. If the presence of a voltage-dependent gate in series with the potassium channel is postulated both for surface and tubular membrane, then Fig. 3 will not represent correctly the current flowing through the tubular membrane. The surface current at any potential would depend on the degree of inactivation of the gate at the conditioning potential. From a qualitative point of view, to obtain the tubular current alone we should subtract less than 15 % of the instantaneous current due to the conditioning steps. This correction will reduce the shift of the zero-current potential and also the amount of relative conductance decrease in the experimental points in Fig. 14. The presence of an access resistance at the mouth of the tubules (Adrian & Peachey, 1973) has not been considered in our model. This resistance would tend to reduce non-uniformities in the tubular lumen. In such a case the relation between relative conductance and zero-current potential that would be theoretically expected would lie very close to curve B in Fig. 6, increasing the disagreement between experimental and computed results. The disagreement depends also on the value of the part of the total current which
E. MIANCINELLI AND A. PERES is estimated to flow through the surface membrane at the beginning of the conditioning steps. The higher this value the greater the disagreement. The value of 15 % that we used is already a lower limit, compared to the estimates of Almers (1972b) and of Eisenberg & Gage (1969). The results of the present paper confirm the observation made by Almers (1972b) that the conductance changes after hyperpolarization cannot be wholly due to depletion, the further reduction being imputable to a voltage-dependent permeability change. Studying the recovery of conductance after a depleting pulse, Almers was able to construct the steady-state activation curve for this gate. His Fig. 5 shows the shape of this curve (Pa,) which he found to be equal to 1 (fully open gate) at resting potential and to begin to decrease with hyperpolarizations beyond 120 mV. While our experiments confirm the existence of a voltage-dependent gate, we are unable to estimate the shape of its activation curve. In fact only if the gate were not present in the surface membrane would we be able to obtain POL as the ratio between the experimental and the computed points in Fig. 14A. It is apparent in Fig. 14 that the difference between experimental and computed results is already significant at values of conditioning potential more positive than - 120 mV. Though this could be due to the model, our results suggest that the position of the PO,, curve on the voltage axis might be shifted a little to the right with respect to the curve given by Almers (1972b). 318
-
The authors would like to thank Dr R. H. Adrian for helpful criticism, Professor A. Ferroni and Dr F. Andrietti for reading the manuscript and the Centro Ricerca di Automatica ENEL for assistance in the computations. This work has been partially supported by a CNR research grant, No. 78.02099.04. REFERENCES
ADRIAN, R. H. (1956). The effect of internal and external potassium concentration on the membrane potential of frog muscle. J. Physiol. 133, 631-658. ADRIAN, R. H., CHANDLER, W. K. & HODGKIN, A. L. (1969). The kinetics of mechanical activation in frog muscle. J. Physiol. 204, 207-230. ADRIAN, R. H., CHANDLER, W. K. & HODGKIN, A. L. (1970a). Voltage clamp experiments in striated muscle fibres. J. Physiol. 208, 607-644. ADRIAN, R. H., CHANDLER, W. K. & HODGKIN, A. L. (1970b). Slow changes in potassium permeability in skeletal muscle. J. Physiol. 208, 645-668. ADRIAN, R. H. & FREYGANG, W. H. (1962). The potassium and chloride conductances of frog muscle. J. Physiol. 163, 61-103. ADRIAN, R. H. & PEACHEY, L. D. (1973). Reconstruction of the action potential of frog sartorius muscle. J. Physiol. 235, 103-131. ALMERS, W. (1972a). Potassium conductance changes in skeletal muscle and the potassium concentration in the transverse tubules. J. Physiol. 225, 33-56. ALMERS, WV. (1972b). The decline of potassium permeability during extreme hyperpolarization in frog skeletal muscle. J. Physiol. 225, 57-83. BARRY, P. H. & ADRIAN, R. H. (1973). Slow conductance changes due to potassium depletion in the transverse tubules of frog muscle fibres during hyperpolarizing pulses. J. Membrane Biol. 14, 243-292. EISENBERG, R. S. & GAGE, P. W. (1969). Ionic conductances of the surface and transverse tubular membrane of frog sartorius fibres. J. gen. Physiol. 53, 279-297. HODGKIN, A. L. & HOROWICZ, P. (1960). The effect of sudden changes in ionic concentrations on the membrane potential of single muscle fibres. J. Physiol. 153, 370-385. HODGKIN, A. L. & NAKAJIMA,S. (1972). The effect of diameter on the electrical constants of frog skeletal muscle fibres. J. Physiol. 221, 105-120. NAKAJIMA, S., NAKAJIMA, Y. & PEACIEY, L. D. (1969). Speed of repolarization and morphology of glycerol-treated muscle fibres. J. Phy.siol. 200, 115-1 16P.
