LECTURE 4: MEASURING MEMBRANE CONDUCTANCE AND CAPACITANCE &VOLTAGE-CLAMP RECORDING

REQUIRED READING: Kandel text, Chapters 8, 9 (beginning), pgs 140-153

We have talked about the properties of solid-state RC electrical circuits.

We have also learned that the resting membrane consists of electrical components:The membrane is a capacitor

Channels create ion-specific conductancesConcentration gradients establish ion-specific battery potentials

We show here that a cell at rest exposed to a transition in transmembrane voltage responds as a simple RC circuit

SO LONG AS THE VOLTAGE CHANGES DO NOT ALTER THE OPEN/CLOSED STATES OF ANY MEMBRANE CHANNELS !!!

In an RC electrical circuit, we can measure the resistance (conductance) andcapacitance of components by analyzing currents and component voltages

induced by applying a voltage step.

RB

RA +

- VBat

SWITCH CLOSED t = 0 sec

C

IC

IA

IB

time 0

I A

time 0

I A

IA=10 mV/(RA + RB)

time 0

I A

IC

C= QC / VC = QC (RA + RB) /10 mV RB

WHEN RA <<< RB IA=10mV/RBC = QC / 10mV

AND

= 10 mV

GRAPHIC AND CIRCUIT REPRESENTATIONS OF ION FLOWSACROSS THE MEMBRANE AT THE RESTING POTENTIAL

inin

outout

K+

K+

+ + +

- - -

K+

K+

K+

K+

K+

K+ Na+

Na+

+ + ++ + ++ + +

- - - - - -- - - - - -- - -

+ + + + + +Vm = - 71 mV

EEKK = - 82 mV = - 82 mV EENaNa = + 85 mV = + 85 mV

IIK K = = - - IINaNa

AT STEADY STATE:

inin

outout

+

-EK = - 82 mV

gK = 2 nS

I K =

2

2

pA

RK = 0.5 G

inin

outout

-+ENa = + 85 mV

gNa = 0.14 nS

I Na =

-

22

p

A

RNa = 7.1 G

EEKK + I + IKKRRKK = Vm = EENaNa + I + INaNaRRNaNa-82 mV + (22 pA)(0.5 G-82 mV + (22 pA)(0.5 G)) = -71 mV = +85 mV + (-22 pA)(7.1 G+85 mV + (-22 pA)(7.1 G))

Vminin

outout

= -71 mV+++

-- -

GRAPHIC AND CIRCUIT REPRESENTATIONS OF ION FLOWSACROSS THE MEMBRANE AT THE RESTING POTENTIAL

inin

outout

K+

K+

+ + +

- - -

K+

K+

K+

K+

K+

K+ Na+

Na+

+ + ++ + ++ + +

- - - - - -- - - - - -- - -

+ + + + + +Vm = - 71 mV

EEKK = - 82 mV = - 82 mV EENaNa = + 85 mV = + 85 mV

IIK K = = - - IINaNa

AT STEADY STATE:

EErestrest + I + IleakleakRRleakleak = Vm -71 mV + (0 pA)(0.48 G-71 mV + (0 pA)(0.48 G)) = -71 mV

inin

outout

+

-Erest = - 71 mV

gleak = 2.14 nS

I leak =

0

p

A

Rleak = 0.48 G

Vminin

outout

= -71 mV+++

-- -

Vcommand

Ileak

Erest

slope = gleak

(Vcommand - Erest ) x gleak = Ileak

VOLTAGE STEP, TOTAL CHANNEL CONDUCTANCE, ANDLEAK CURRENT OBEY OHM’S LAW

For simplicity, we can combine all of thechannels and gradients contributing to

resting potential into one circuit containing:

a resting potential battery

Erest

and the total conductance of all channels

gtotal

gleak

Vc

om

man

d

Cmem +-

out

in

Erest

+-

Ileak

A VOLTAGE CHANGE ACROSS THE MEMBRANE ACTS INDEPENDENTLYON EACH COMPONENT OF THE MEMBRANE CIRCUIT

inin

outout

+

-EK = - 82 mV

gK = 2 nS

I K =

2

2

pA

RK = 0.5 G

inin

outout

-+ENa = + 85 mV

gNa = 0.14 nS

I Na =

-

22

p

A

RNa = 7.1 G

- -Vm =

inin

outout

-71 mV

+++

-+-

- 61 mV

IMPOSEIMPOSECOMMANDCOMMANDVOLTAGEVOLTAGE

10 mV10 mVABOVEABOVE

RESTINGRESTINGPOTENTIALPOTENTIAL

inin

outout

+

-EK = - 82 mV

gK = 2 nS

I K =

4

2

pA

RK = 0.5 G

inin

outout

-+ENa = + 85 mV

gNa = 0.14 nS

I Na =

-2

0.6

pA

RNa = 7.1 G

- -Vm =

inin

outout

-61 mV

++

+-

- 61 mV

inin

outout

+

-EK = - 82 mV

gK = 2 nS

I K =

4

2

pA

RK = 0.5 G

inin

outout

-+ENa = + 85 mV

gNa = 0.14 nS

I Na =

-2

0.6

pA

RNa = 7.1 G

- -Vm =

inin

outout

-61 mV

++

+-

- 61 mV

VOLTAGE STEP 10 mV

TOTAL CHANNEL CONDUCTANCE

gtotal(leak) 2.14 nS

NET STEADY STATECURRENT FLOW

Ileak 21.4 pA

DISCHARGE = 10 mV x C

V

g

I

x

=

VOLTAGE STEP FROM RESTING POTENTIAL INDUCES CAPACITANCETRANSIENT CURRENT AND STEADY-STATE LEAK CURRENT

time 0

I TO

TA

L

Ileak = (Vcommand - Vrest ) x gtotal(leak)

IC

0

x

=

EFFECT OF VOLTAGE STEP ON CURRENTS AND VOLTAGEAT A DIFFERENT SITE WITHIN NEURON

Rmem

+

-

Vc

om

- E

res

t

Cmem Rmem Cmem

out

in Raxial

SITE OF COMMAND DIFFERENT SITE

time 0

I TO

TA

L

Ileak = (Vcommand - Erest ) / Rmem

IC

0

time 0

I TO

TA

L

Ileak = (Vcommand - Erest )/

IC

0 (Rmem + Raxial )

If Raxial is significant, Ileak and voltage divergence from Erest at different site

is less and voltage divergence is delayed by Raxial x C time constant

DETERMINANTS OF AXIAL RESISTANCE

Raxial ~ Distanceaxial / Areacross-sectional

Cell soma has relatively large diameter (3 - 30 microns) and cross-sectional area,

so Raxial in soma is negligible. Therefore, transmembrane voltage will always be the same at all points around the soma, even during rapid current/voltage changes.

Raxial is significant along the axon and thin dendrites. The narrower an axon’s

diameter, the larger is Raxial, and the greater delay and attenuation of a voltage change occuring at a distance within the cell.

THE IDEAL VOLTAGE CLAMP

Voltage clamp is the ability to rapidly and stably fix a voltage difference between 2 points.

When used in conjunction with a whole-cell patch, voltage clamp allows forthe immediate and stable shift in the voltage across the cell membrane.

Voltage clamp allows for the measurement of passive membrane properties(leak conductance and membrane capacitance)

along with voltage- and time-dependent changes in ion-specific conductances

The ideal voltage clamp can be simulated as a “command” voltage battery connected to an on/off switch

Rleak

Vc

lam

p

Cmem +-

out

in

Erest

+- +-

+-

Vclamp

patchpipet

CELL

bath(grounded)

THE REAL VOLTAGE CLAMP

A real voltage clamp consists of a feedback amplifier which continuously compares the voltage across the membrane to a command voltage, and injects

sufficient current into the cell to make this voltage difference = 0

VB

VA

IC

C

B

A

SIMPLIFIED SCHEMATICOF A TRANSISTOR

AMPLIFIER

IICC ~~ VVBB -- VVAA +-patchpipet

CELL

bath(grounded)

Vmembrane

VcommandPOWERPOWERSOURCESOURCE

Iinject

IinjectCURRENTCURRENTMONITORMONITOR

Icap

Imem

ground

ground

If any changes occur in membrane channelscausing new currents and drift of Vm,

the voltage clamp very rapidly senses thisdrift and adjusts current injection to

maintain Vm = Vcommand

REAL VOLTAGE CLAMP ANALOGOUS TO“WATER LEVEL CONTROLLER” IN LEAKY TUB

Inside of tub (inside cell) has width and depth (capacitance) and has an open drain (leak conductance, gleak). Baseline water level (resting potential, Vrest) is set by water level (resting battery, Erest) outside the tub.

The water level controller (voltage clamp) measures water level in tub (Vmembrane) and compares it to an adjustable water level set value (Vcommand) and then injects or sucks water from the tub (current injection) with a pressure proportional to the difference in levels (Vcommand - Vmembrane).

When a new water level command is applied, the system first injects/sucks a large amount of water to reset water level (Iinject = IC) and then continues to inject/suck smaller amount of water to compensate for water passing through drain and thereby maintains command level (Iinject = Ileak). The flow of water through drain obeys “Ohm’s law”, determined by how much command level differs from resting level and by size of drain. Ileak = (Vcommand - Erest) x gleak

Next Lecture: ION CHANNELS: STRUCTURES AND FUNCTIONS

REQUIRED READING: Kandel text, Chapters 6,9