Biomembrane Electrochemistry - American Chemical Society

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19 Determination of Potassium- and Sodium-Channel Relaxation Times in Squid Nerve Fibers from Membrane Admittance Analysis Harvey M . Fishman and H. Richard Leuchtag 1

2

Department of Physiology and Biophysics, University of Texas Medical Branch, Galveston, TX 77555-0641 Department of Biology, Texas Southern University, Houston, TX 77004 1

2

Complex admittances in the 12.5-5000-Hz frequency range were acquired rapidly (80 ms) and at various times after step voltage clamps of squid giant axons. In a Κ -conducting membrane (Na conduction blocked) at 12.5 °C, admittances were in a steady state 100 ms after steps, whereas in a Na -conducting membrane (Κ conduc tion blocked) at 8 °C, admittances were time-invariant in the interval from 20 ms to 1 s after step changes. Admittances determined in the —65- to 0-mV voltage range were fitted by an admittance model to obtain conduction relaxation times for Κ and for Na as a function of voltage. Evaluation of macroscopic conduction in membranes via rapid admittance determinations provides a direct linear analysis that relates to linear theory and to Markovian models of single-channel conduction processes. +

+

+

+

+

+

C (ONDUCTION OF M I PULSES along single giant nerve

fibers o f the s q u i d is

p o w e r e d b y two energy sources: a potassium i o n gradient that maintains the m e m b r a n e i n an excitable (resting) state, w i t h the axon interior negative (about —65 m V ) , a n d a sodium i o n gradient that, w h e n N a

+

conduction is

triggered, transiently changes the polarity o f the axon interior to a positive value a n d thereby generates an electrical p u l s e — t h e action potential ( I ) . T h e 0065-2393/94/0235-0415 $08.00/0 © 1994 American Chemical Society

Blank and Vodyanoy; Biomembrane Electrochemistry Advances in Chemistry; American Chemical Society: Washington, DC, 1994.

416

BIOMEMBRANE ELECTROCHEMISTRY

i o n flows are controlled b y two i o n - c o n d u c t i n g systems i n the m e m b r a n e : the Κ system a n d t h e N a system. T h e conductance o f the N a system has a remarkably steep dependence o n voltage over a range o f voltages just above the resting potential, a n d depolarization o f the m e m b r a n e triggers the onset o f the action potential by allowing N a to flow into the axon. +


The

i o n - c o n d u c t i n g systems

are based

o n t w o specialized macro-

molecules, t h e Κ channel (2) a n d t h e N a channel (3), e m b e d d e d i n a relatively inert p h o s p h o l i p i d bilayer. T h e s e channels differ i n i o n selectivity, kinetics, a n d pharmacological sensitivities. H o w e v e r , b o t h channels have at least two states, one c o n d u c t i n g a n d the other nonconducting, a n d undergo stochastic transitions between states, t h e probabilities o f w h i c h d e p e n d o n membrane voltage (present a n d recent past) a n d temperature. I n cells that allow direct access t o the plasma membrane, unitary currents measured b y micropatch techniques reflect transitions to a c o n d u c t i n g state (openings) and transitions to a n o n c o n d u c t i n g state (closings) (4). T h e smooth currents measured b y standard (nonmicropatch) methods represent sums o f m a n y unitary currents ( 5 ) . I n microkinetic analyses o f i o n c o n d u c t i o n through these single channels, transitions between channel n o n c o n d u c t i n g a n d c o n d u c t i n g states are usually assumed t o b e described b y a M a r k o v i a n process. U n d e r this assumption a comparable macrokinetic description o f ion-channel-conduction relaxation times is available directly f r o m a population o f identical (single) channels. T h i s analysis is accomplished b y fitting a linear m o d e l o f m e m b r a n e drivingpoint functions (complex i m p e d a n c e or admittance) to low-frequency ( 1 - 5 0 0 0 H z ) admittance determinations made i n a steady state d u r i n g voltage steps (clamps) o f the m e m b r a n e potential. W i t h the advent o f methods for r a p i d determination o f the " s m a l l - s i g n a l " admittance o f membranes (6-9), a direct linear analysis o f i o n c o n d u c t i o n i n membranes became a viable alternative (8, 10, 11) t o conventional voltage c l a m p analysis, w h i c h was shown recently to y i e l d results that are not equivalent to a linear analysis (12). A n expression for the complex admittance, Y(jf), o f an axon membrane is obtained b y linearizing the H o d g k i n - H u x l e y ( H H ) equations ( I ) a n d b y applying a L a p l a c e transformation (13, 14). T h e m e m b r a n e admittance is then given b y the general expression

* ( / > - ^ +* + Σ

(

1

+

*.

Λ > )

(»)

where j = )/ — 1 , / is frequency, C is capacitance, g is a frequency-inde pendent i o n conductance, a n d t h e g ( l 4- j2Trfr )~ terms are t h e fre quency-domain counterparts o f t i m e - d o m a i n relaxations o f the i o n conduc tances associated w i t h activation a n d inactivation kinetics o f the specific type o f i o n channel b e i n g described. {

p

?

l


19.

FISHMAN AND LEUCHTAG

Relaxation Times from Admittance Analysis

417


D e s c r i p t i o n o f ion-channel kinetics via admittance analysis provides a framework w i t h i n w h i c h linear kinetic models can be c o m p a r e d to macro scopic data (from a p o p u l a t i o n o f channels) i n a m e m b r a n e . Analysis o f c o n d u c t i o n v i a driving-point-function determinations also provides p r o p e r data (from a true linear analysis) for comparison w i t h the relaxation times obtained f r o m microscopic data f r o m one or a small n u m b e r o f channels i n a m e m b r a n e patch isolated b y a micropipette (4). I n M a r k o v m o d e l i n g , the o p e n - a n d closed-time distributions are fitted to sums o f exponential func tions (15).

Materials and Methods Preparation and Solutions. G i a n t axons f r o m the hindmost stellar nerve o f squid, Loligo pealei, were used i n these experiments. C o n v e n t i o n a l internal axial-electrode techniques, as described previously (10), were i m p l e m e n t e d . Axons were p l a c e d i n a cooled, flowing external artificial seawater ( A S W ) composed of 4 3 0 - m M N a C l , 1 0 - m M C a C l , 5 0 - m M M g C l , 2 0 - m M sucrose, a n d 5 - m M tris(hydroxymethyl)aminomethane hydrochloride b u f f e r e d to p H 7.4 at 22 °C. T o study K conduction, intact axons (not internally perfused) were used a n d l - μ Μ tetrodotoxin ( T T X ) was a d d e d to the A S W to eliminate N a c o n d u c t i o n . F o r studies o f N a conduction, the A S W c o n tained 5 0 % o f the N a fisted a n d the r e m a i n i n g 5 0 % was replaced w i t h tetramethylammonium i o n . T h e purpose o f the r e d u c t i o n i n N a concentra tion was to facilitate spatial control o f the m e m b r a n e potential. T h e axons i n Na experiments were internally perfused w i t h a solution c o m p o s e d o f 1 0 0 - m M potassium glutamate, 2 0 0 - m M cesium glutamate, 4 0 - m M C s F , 4 0 0 - m M sucrose, a n d 5 - m M T r i s C l b u f f e r e d to p H 7.4 at 22 °C. I n addition, 1 - m M 3,4-diaminopyridine was a d d e d to the perfusate to assure blockage o f K conduction. 2

2

+

+

+

+

+

+

+

Data Analysis. C o m p l e x admittance determinations w e r e fitted b y an admittance function (13, 14, 16) based o n the linearized H H equations (J). A d m i t t a n c e measurements were made u n d e r steady-state conditions (see Figures 2 a n d 4). Series resistance (R ), the access resistance between the two voltage electrodes a n d u p to the i n n e r a n d outer surfaces o f the axon m e m b r a n e was not r e m o v e d f r o m measurements. Instead R was i n c l u d e d and d e t e r m i n e d i n the fit o f the steady-state admittance m o d e l to the data. T h e measured complex admittance, therefore, is s

s



418

T h e membrane admittance, Y (jf), N a conduction, was m o d e l e d b y

d u e to capacitance, leakage, a n d K

m

+

and

+

Ujf)

= 0'2W) c a

m

+

(3)

+ y (jf) + Y (iD

g L

K

Ne


where the admittance o f the potassium system is

a n d the admittance o f the sodium system is

Wif) = g „ +

r r $ ^

+

Na

( 5 )

C is m e m b r a n e capacitance, α is a n u m b e r less than 1 that reflects the constant phase angle character o f C ( 1 7 ) , g is the leakage conductance, and g and g are the c h o r d conductances o f the respective K a n d N a systems. T h e g t e r m reflects the kinetic component (activation) o f the potassium conductance that relaxes w i t h t i m e constant τ a n d the g a n d g terms reflect the kinetic components (activation a n d inactivation, respec tively) o f the sodium conductance that relax w i t h time constants T a n d τ , respectively. T h e F O R T R A N p r o g r a m C P X F I T was used to fit eqs 2 a n d 3 to complex admittance data obtained o n either a Κ - o r a N a - c o n d u c t i n g m e m b r a n e . m

L

m

K o o

+

N a 0 0

+

n

η

m

h

Λ

m

+

+

C P X F I T e m p l o y e d a g r i d search [ G R I D L S (18)] m e t h o d o f least squares to explore the parameter space o f the frequency f u n c t i o n i n these equations. T h e parameter values that y i e l d e d the best fit ( m i n i m u m chi-square ( χ ) value) was obtained b y m i n i m i z i n g the χ o f the vector difference ( f o r m e d b y the real a n d imaginary parts o f the admittance) between the m o d e l a n d the data at each frequency. A useful feature o f the p r o g r a m was a probe f o r a local m i n i m u m i n the following way. U p o n approach to a m i n i m u m χ , the incremental step size o f each parameter was reset to 1 0 % o f its value a n d fitting resumed. T h i s perturbation i n step size usually p r o d u c e d sufficient change i n the parameter space to p r o c e e d to a deeper m i n i m u m χ i f the p r i o r one was only a local m i n i m u m . 2

2

2

2

Rapid Complex Admittance Measurements. T h e voltage c l a m p system, chamber, a n d axial electrode techniques were described previously (10, 12). B y superposing a repetitive small-amplitude ( 1 - m V root m e a n square) Fourier-synthesized signal (8, 16, 19) onto large step clamps, a current response was a c q u i r e d d u r i n g voltage c l a m p pulses. Immediately


19.


FISHMAN A N D L E U C H T A G

419

after a response was acquired, a 400-point complex admittance was c o m p u t e d (by software) i n the 1 2 . 5 - 5 0 0 0 - H z

frequency range as the ratio o f the

F o u r i e r - t r a n s f o r m e d current response to that o f the smaU-amplitude voltage signal previously stored. F i g u r e 1 shows the t i m i n g o f the acquisition o f admittance data d u r i n g the current response o f a K - c o n d u c t i n g axon m e m b r a n e to a rectangular +

voltage-clamp pulse. T h e horizontal dashed Une w i t h p e r i o d i c vertical marks indicates the start o f a continuously a p p l i e d repetitive 1 - m V signal. A large


voltage step (30 m V ) , f r o m

—65 to —35 m V , was a p p l i e d so that t h e

response to the small synthesized signal o c c u r r e d after a specified p r e m e a surement interval (200 ms i n this example) a n d so that one complete cycle o f the signal ( d u r i n g the data acquisition w i n d o w i n F i g u r e 1) began precisely after the premeasurement interval. T h e small amplitude o f the synthesized signal was necessary to assure that the response that contained the admit tance i n f o r m a t i o n reflected p r i m a r i l y the linear properties o f the m e m b r a n e

(20).

-45mV PREMEASUREMENT INTERVAL 200msec

DATA ACQUIRED 80msec

86-43

Figure 1. Illustration of the acquisition of a K current response during a voltage-clamp step (not shown) and with a 1-mV (root mean square) Fouriersynthesized signal applied continuously and periodically, as indicated by the ticks on the dashed horizontal line. The Κ current response to the small voltage perturbation was acquired after a "premeasurement interval" (200 ms) from the onset of a large voltage step from —65 to — 35 mV. The Fourier transform of the current response together with the previously stored Fourier transform of the synthesized voltage signal enabled calculation of the complex admittance, Y(jf) given by eqs 2, 3, and 4 with Y (]f) = 0, during the time indicated by the dashed box (data acquired) after the step change in membrane voltage. (Reproduced with permission from reference 12. Copyright 1991). +

+

Na


420


T h e premeasurement

interval allowed the m e m b r a n e

to respond to

several cycles o f the perturbation signal before the data were acquired. T h i s condition enabled satisfaction o f the additional requirement that data be obtained i n a steady state so that the frequency structure o f the admittance w o u l d not d e p e n d o n the t i m e o f measurement a n d thus be interprétable i n terms

o f linear time-invariant processes. I n this respect, the

frequency

components i n t r o d u c e d into the admittance function b y the slowly decaying phase o f the mean-current response i n F i g u r e 1 were also e l i m i n a t e d i n the


time d o m a i n . T h i s e l i m i n a t i o n was accomplished b y a coherence e l i m i n a t i o n m e t h o d described previously (8) i n w h i c h a pair o f evoked responses was obtained w i t h an inversion (polarity change) o f the superposed synthesized signal between acquisition o f the p a i r e d responses. T i m e - d o m a i n subtraction o f the pair o f responses e l i m i n a t e d the decaying phase i n the mean current, w h i c h was c o m m o n to b o t h responses. T h e subtraction o f p a i r e d responses also resulted i n reinforcement o f the admittance-containing current response because o f the intentional inversion o f the small voltage signal between response pairs. F i g u r e 2 illustrates acquisition intervals i n a N a - e o n d u e t i n g axon m e m +

brane d u r i n g a rectangular voltage-clamp pulse. T h e crosshatching o n indicates the superposed signal, a n d the crosshatching o n l

m

current response,

w h i c h contains

the admittance

V

m

represents the

information. T h e

dark

horizontal bars are data acquisition periods (80 ms i n duration) after three premeasurement intervals o f 20, 100, a n d 200 ms. Admittances at the same m e m b r a n e potential were d e t e r m i n e d at these times after steps to establish that the m e m b r a n e admittance was invariant a n d thus was i n a steady state 20 ms after a step (see

F i g u r e 5).

0

1

L_l

msec

1

100

m

s

e

1

1

200

c

I « I I I I I I I

— )

I

—I

1

1

1 I !

1

300 1

L_J I 1 1

)

I

Figure 2. Representative drawing of the N a current (l ) response to a voltage-clamp step, shown as V , with superposed synthesized small perturbatio (crosshatching). The acquisition of the current response to the small voltage perturbation, from which the admittance Y(jf) was calculated from eqs 2, 3, and 5 with Y ( j f ) = 0, occurred in the three intervals (20-100, 100-180, and 200-280 ms) marked by heavy horizontal lines under the time scale. +

m

m

K


19.

FISHMAN A N D L E U C H T A G


421

A d m i t t a n c e data are plotted i n the following two ways: 1. A s magnitude,

a n d phase,

= arctan


AY(jf)

B(f) Gif)

functions o f frequency. 2. A s the reciprocal admittance i n the complex impedance plane [X(f)

versus

R(f)],

Z(jf)=R(f)+jX(f)

where Y(jf)

= Gif)

=

1

njf)

+jB{f).

Results K

+

Conduction: Determination of Y ( j f ) i n a Steady State. K

R a p i d admittance determinations i n biological membranes c a n b e used to characterize the linear response o f m e m b r a n e c o n d u c t i o n (8-12, 16) p r o v i d e d that (1) the amplitude o f the voltage stimulus elicits an insignificant nonlinear response (measured as higher harmonics) a n d (2) the admittance determination is invariant w i t h t i m e . W i t h respect to provision 1, the a m p l i tude o f the voltage perturbation used to obtain admittance data at any given m e m b r a n e potential i n these experiments was chosen to b e 1 m V (root m e a n square), w h i c h was shown previously (20) b y h a r m o n i c analysis to y i e l d essentially a linear response i n the s q u i d axon. T o satisfy criterion 2, admit tance data i n a predominantly K - c o n d u c t i n g axon were acquired at 0.1 a n d at 0.5 s after step voltage clamps to membrane potentials ranging f r o m — 80 to 10 m V . These data are shown i n F i g u r e 3, i n w h i c h the reciprocal o f the admittance is p l o t t e d as 400 frequency points i n the complex i m p e d a n c e plane. T h e data were obtained f r o m the same axon at each o f the two times after steps to the same i n d i c a t e d voltages. T h e impedance locus at the t w o times shows slight differences at m e m b r a n e voltages greater than —60 m V . Nevertheless, best fits o f eqs 2, 3, a n d 4 w i t h Y (jf) = 0 (solid curves) to these data y i e l d e d estimates o f the relaxation time, τ , that were nearly identical (Table I). C o m p a r a b l e results were obtained i n eight other axons. T h u s , f o r a K - c o n d u c t i n g axon the m e m b r a n e admittance appears to b e +

Na

η

+


422



-70mV

-50mV

o.isec-y ) :

sec

-45mV

0.1 sec-*'

^

0.5sec

)

.8

1.6

-!5mV

R(f)Figure 3. Admittance data from a Κ ^-conducting membrane and curve fits (solid curves) of eqs 2 , 3, and 4 with Yw (jf) = 0 plotted in the complex plane [X(0 vs. R(f)] as impedance [Z(jf) = R(f) + jX(f) = Y f j f j ] loci (400 frequency points) over the 12.5-5000-Hz frequency range. These data were acquired rapidly as complex admittance data, as illustrated in Figure 1, at premeasurement intervals of 0.1 and 0.5 s after step voltage clamps to each of the indicated membrane potentials from a holding of —65 mV. The near superposition and similarity in shape of the two loci at 0.1 and 0.5 s, at each voltage, indicates that the admittance data reflect a steady state in this interval after step clamps. Axon 86-41 internally perfused with buffered KF and externally perfused in ASW + TTX at 12 °C. The membrane area is 0.045 cm . e

_ J

2


19.

FlSHMAN AND


LEUCHTAG


T a b l e I. Estimates o f τ

423

η

V(mV)

0.1s

0.5 s

-75 -70 -60 -55 -50 -45 -35 -25 -15 -5 10

7.3 10.1 12.7 11.0 8.3 5.0 2.1 1.0 0.84 0.62 0.86

6.9 9.8 12.1 10.9 7.0 5.3 2.0 1.2 0.86 0.71 0.67

N O T E : All values are given in milliseconds and are taken from fits of the admittance data plotted as impedance loci in Figure 3 at the two premeasurement intervals (PMI) 0.1 and 0.5 s after step changes to membrane voltage, V.

sufficiently time-invariant i n the interval between 0.1 a n d 0.5 s after a step to consider measurements i n this interval to reflect a steady-state c o n d i t i o n .

Evaluation of τ ( V ) from Admittance Analysis. η

a plot o f the estimates of τ

η

F i g u r e 4 shows

obtained f r o m m o d e l curve fits to admittances

acquired 200 ms after steps a n d obtained at nine m e m b r a n e voltages span n i n g the range f r o m —65 to —25 m V i n a K - c o n d u c t i n g axon. T h e three +

data points p l o t t e d at each m e m b r a n e voltage i n F i g u r e 4 w e r e obtained i n the following manner. A t each m e m b r a n e potential, eight separate admit tance determinations were made. A n averaged admittance was calculated b y averaging the real a n d imaginary parts f r o m the eight single determinations. A t each frequency the real a n d imaginary parts ( f r o m each o f the single admittance determinations that went into the average calculation) w e r e used to calculate the standard deviation o f the real a n d imaginary parts f r o m the corresponding parts o f the averaged admittance. T w o n e w admittance func tions were then generated b y adding (and subtracting) one standard deviation to (from) the real a n d imaginary parts o f the averaged admittance at each frequency. Best fits o f eqs 2, 3, a n d 4 w i t h Y (jf) Na

= 0 to these three

admittance functions at different voltages y i e l d e d the values p l o t t e d as A V E , + S D , a n d — S D i n F i g u r e 4. Because the data fits involved a l l 400 points, the A V E points d i d not always fall between the — S D a n d + S D points. T h e appearance o f the admittance-derived T (V) n

curve is qualitatively

similar to that p r o d u c e d b y a Η Η analysis o f large step-voltage H o w e v e r , a quantitative comparison (12)

responses.

showed significant discrepancies

between the values at corresponding potentials a n d variability i n the discrep-


424


Figure 4. Estimates of the potassium-con ductance relaxation time, τ , from fits of eqs 2, 3, and 4 with Y — 0 to admit tance determinations at various mem brane voltages, similar to those shown in Figure 3. Filled triangles are from fits of the average (AVE) of the real and imagi nary parts of eight separate, successive admittance determinations at each volt age. Open circles and squares are from fits of 1 standard deviation added to ( + SD) or subtracted from ( — SD) the real and imaginary parts of the AVE admittance. Intact axon 87-39 in ASW + TTX(lpM)at 12.5 °C. η


Na

0

I

-80

I

I

I

I

-60

I

-40

I -20

mV

I

I 0

V

m

ancies at different voltages. T h u s the functional shapes o f T ( V ) o b t a i n e d n

f r o m the same axon b y a Η Η analysis a n d b y an admittance analysis are different.

Na State.

Conduction: Determination of Y (j[f) in a Steady

+

Na

A d m i t t a n c e data for a p r e d o m i n a n t l y N a - c o n d u c t i n g m e m b r a n e , +

a c q u i r e d after steps to four m e m b r a n e voltages i n the - 6 0 - to 0 - m V range, are p l o t t e d as magnitude a n d phase angle functions o f frequency i n F i g u r e 5 . A t each voltage, data were obtained at the premeasurement intervals (20, 100, a n d 200 ms) shown i n F i g u r e 2 a n d d e s c r i b e d therein. T h e most obvious change i n the admittance f u n c t i o n w i t h depolarization is the low-frequency asymptotic behavior o f the phase f u n c t i o n . A t — 60 m V (near the open-circuit rest potential) the low-frequency e n d o f the phase f u n c t i o n tends t o w a r d 0°, whereas at d e p o l a r i z e d potentials ( — 40, — 20, a n d 0 m V ) the phase exceeds 180°, w h i c h is the manifestation o f a steady-state negative conductance p r o d u c e d b y the N a - c o n d u c t i n g system, as described previously ( 1 0 ) . T h e +

180° phase angle at depolarized voltages reverted to an angle that approached 0°, as at —60 m V , at a l l voltages after external application o f 1 - μ Μ T T X . N o significant differences w e r e seen i n these data a c q u i r e d at different times after step changes to the same voltage, a n d thus the admittances d e t e r m i n e d 20 ms after steps to a N a - c o n d u c t i n g m e m b r a n e reflected a steady-state +

condition.

Best Fits of Impedance Data with the Y ( jf ) Model. Na

the existence

o f a steady-state

After

c o n d i t i o n 20 ms after step clamps was

established, w e a c q u i r e d a set o f admittance data i n the 5 - 2 0 0 0 - H z fre quency range i n a N a - c o n d u c t i n g axon at a premeasurement interval o f 100 +



19.

FISHMAN AND LEUCHTAG

425

Relaxation Times from Admittance Andy sis

Hz

f

Hz

f

Figure 5. Admittance data plotted as magnitude and phase angle vs. frequency as determined at the three premeasurement intervals (20, 100, and 200 ms) shown in Figure 2 and at the indicated membrane voltages. The superposition of the admittance data at each voltage indicates that the admittance is timeinvariant in the interval from 20 to 200 ms after step changes in membrane voltage. Axon 87-19 internally perfused with the perfusate described in the text and externally perfused with ASW at 8 °C.

ms. These data (frequency points) are p l o t t e d i n the complex impedance plane i n F i g u r e 6 at eight m e m b r a n e voltages ranging f r o m —65 to 0 m V . T h e solid curves are loci o f the best fits o f the reciprocal o f the admittance function defined b y eqs 2, 3, a n d 5, w i t h Y (jf) K

= 0. T h e striking feature o f

the loci i n the complex impedance plane is the change f r o m a right half-plane [R(f)

> 0] locus at

[R(f)

< 0] locus at depolarized m e m b r a n e voltages. T h e impedance

= R(f)

+jX(f)]

tance [R(f)

—65

m V (near rest potential) to a left half-plane [Z(jf)

behavior at depolarized voltages reflects a negative resis

< 0] process that is characteristic o f the N a - c o n d u c t i n g system +

o f the axon m e m b r a n e .

Evaluation of T (V) and T (V) from Admittance Analysis. to

h

A p p l i c a t i o n o f the averaged admittance methodology to a N a - c o n d u c t i n g +

axon enabled us to obtain three data points at each m e m b r a n e voltage. F i g u r e 7 shows these data, w h i c h w e r e obtained f r o m best fits o f eqs 2, 3, a n d 5 w i t h


426


55mV

-65mV


X(f)

0

Figure 6. Admittance data from a No.*-conducting membrane and curve fits (solid curves) of eqs 2, 3, and 5 with Y ()f) — 0 plotted in the complex plane [X(f) vs. R(f)] as impedance loci (400 frequency points) over the frequency range 5 to 2000 Hz. Same axon and conditions as in Figure 5. K

Y ( j f ) = 0. T h e dependence o f r a n d i o n voltage has the appearance o f time constants d e r i v e d f r o m a H H analysis. H o w e v e r , as w i t h the Κ ^ e o n ducting system, a quantitative comparison indicated discrepancies i n the estimates obtained f r o m a H H analysis w i t h respect to estimates obtained f r o m a n admittance analysis at all voltages ( 1 2 ) . K

m

h

Discussion T h e complex admittance m e t h o d described here allows data to b e analyzed without reference to any particular m o d e l . T h i s c o n d i t i o n is particularly important at this t i m e , w h e n n e w data a n d n e w concepts are challenging previously accepted concepts. T h e elucidation o f the p r i m a r y structure o f channel proteins ( 2 , 3 ) has stimulated the development o f a n u m b e r o f structurally oriented models (e.g., references 21 a n d 22). I n addition, n e w physical a n d mathematical concepts have b e e n brought to bear o n the p r o b l e m o f channel gating i n excitable membranes. These concepts i n c l u d e



19.

FISHMAN AND L E U C H T A G


427

Figure 7. Estimates of the sodium-conductance activation relaxation time, r , and inactivation relaxation time, T , from fits of eqs 2, 3, and 5 Y (jf) = 0 to admittance determinations at various membrane voltages, similar to those shown in Figure 6. Filled triangles are fromfitsof the average (AVE) of the real and imaginary parts of eight separate, successive admittance determinations at each voltage. Open circles and squares are from fits of 1 standard deviation added to ( + SO) or subtracted from ( — SD) the real and imaginary parts of the AVE admittance. Axon 87-37 internally perfused with the perfusate described in text and externally perfused with ASW at 9 °C. m

h

K

transitions between ferroelectric a n d superionically c o n d u c t i n g states i n N a channels (23, 24), fractal kinetics (25), a n d chaotic rather than stochastic models (26).

Acknowledgments W o r k supported i n part b y O N R G r a n t N 0 0 0 1 4 - 9 0 - J - 1 1 3 7 a n d b y Texas A d v a n c e d Technology grant 004952011. H . R . L e u c h t a g thanks Texas South ern University for a sabbatical leave a n d the Materials Research Laboratory, Pennsylvania State University, for its hospitality.

References 1. Hodgkin, A . L.; Huxley, A. F. J. Physiol. 1952, 115, 111-222. 2. Takumi, T.; Ohkubo, H.; Nakanishi, S. Science (Washington, D.C.) 1988, 1042-1045.


242,


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ELECTROCHEMISTRY

3. Noda, M.; Ikeda, T.; Suzuki, H.; Yakeshima, H.; Takahashi, T.; Kuno, M.; Numa, S. Nature (London) 1986, 322, 826-828. 4. Hamill, O . P.; Marty, Α.; Neher, Ε.; Sakmann, Β.; Sigworth, F . J. Pflugers Arch. 1981, 391, 85-100. 5. Sigworth, F . J.; Neher, E . Nature (London) 1980, 287, 447-449. 6. Husimi, Y.; Wada, A . Rev. Sci Instrum. 1976, 47, 213-219. 7. Poussart, D.; Ganguly, U. S. Proc. IEEE 1977, 65, 741-747. 8. Fishman, H. M.; Moore, L . E . ; Poussart, D . In The Biophysical Approach to Excitable Systems; Adelman, W . J., Jr.; Goldman, D. E., Eds.; Plenum: New York, 1981; pp 65-95. 9. Kottra, G . ; Fromter, E . Pflugers Arch. 1984, 402, 409-420. 10. Fishman, H. M.; Leuchtag, H. R.; Moore, L. E. Biophys. J. 1983, 43, 293-307. 11. Hayashi, H.; Fishman, H. M. Biophys. J. 1988, 53, 747-757. 12. Fishman, H. M.; Lipicky, R. J. Biophys. Chem. 1991, 39, 177-190. 13. Chandler, W . K.; FitzHugh, R.; Cole, Κ . S. Biophys. J. 1962, 2, 105-127. 14. Mauro, A.; Conti, F.; Dodge, F.; Schor, R. J. Gen. Physiol. 1970, 55, 497-523. 15. Colquhoun, D . ; Hawkes, A . G . In Single-Channel Recording; Sakmann, Β.; Neher, E . , Eds.; Plenum: New York, 1983; pp 135-175. 16. Miyamoto, S.; Fishman, H. M. Ferroelectrics 1988, 86, 129-146. 17. Cole, Κ. S. Membranes, Ions and Impulses; University of California Press: Los Angeles, C A , 1972; pp 42-43. 18. Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969; pp 208-213. 19. Nakamura, H.; Husimi, Y.; Wada, A . Jpn. J. Appl. Phys. 1977, 16, 2301-2302. 20. Moore, L . E.; Fishman, H. M.; Poussait, D . J. Membr. Biol. 1980, 54, 157-164. 21. Guy, H. R.; Seetharamulu, P. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 508-512. 22. Kosower, E. M. FEBS Lett. 1985, 182, 234-242. 23. Leuchtag, H. R. J. Physiol. 1989, 418, 10P. 24. Leuchtag, H. R. 1990 International Symposium on Applications of Ferroelectrics; I E E E : New York, 1991; pp 279-283. 25. Liebovitch, L . S.; Toth, T. I. Ann. Biomed. Eng. 1990, 18, 177-194. 26. Liebovitch, L . S.; Toth, T. I. J. Theor. Biol. 1991, 148, 243-267. RECEIVED

for review January

29, 1991. ACCEPTED

revised manuscript August 4 ,

1992.


Biomembrane Electrochemistry - American Chemical Society

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