# LECTURE 4: MEASURING MEMBRANE CONDUCTANCE AND CAPACITANCE & 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 conductances Concentration 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) and capacitance of components by analyzing currents and component voltages induced by applying a voltage step. IA RA + VBat -= 10 mV IC RB IB IA SWITCH CLOSED t = 0 sec C 0 time IA IA

IC C= QC / VC = QC (RA + RB) /10 mV RB IA=10 mV/(RA + RB) 0 0 time WHEN RA <<< RB IA=10mV/RB AND time C = QC / 10mV GRAPHIC AND CIRCUIT REPRESENTATIONS OF ION FLOWS ACROSS THE MEMBRANE AT THE RESTING POTENTIAL +++ - - - +++ - - - K+ K+ +++ - - - K+ EK = - 82 mV K+ +++ +++ - - -

- - - - - - K+ Na+ K+ ENa = + 85 mV IK = - INa gK = 2 nS RK = 0.5 G +++ out + --- - EK = - 82 mV in Vm = - 71 mV Na+ AT STEADY STATE: out IK = 22 pA in ++ K+ out Vm = -71 mV in

INa = - 22 pA out + K+ gNa = 0.14 nS RNa = 7.1 G in EK + IKRK = Vm = ENa + INaRNa E +Na -82 mV + (22 pA)(0.5 G) = -71 mV = +85 mV + (-22 pA)(7.1 G) = + 85 mV GRAPHIC AND CIRCUIT REPRESENTATIONS OF ION FLOWS ACROSS THE MEMBRANE AT THE RESTING POTENTIAL out + +++ - - - K+ +++ - - - K+ K+ +++ - - - K+ K+

Na+ +++ +++ - - - - - - - - - K+ EK = - 82 mV K+ ENa = + 85 mV IK = - INa out gleak = 2.14 nS Rleak = 0.48 G + Erest = - 71 mV Vm = - 71 mV Na+ AT STEADY STATE: Ileak = 0 pA in ++ K+ +++ out --- -

in in Erest + IleakRleak = Vm -71 mV + (0 pA)(0.48 G) = -71 mV Vm = -71 mV VOLTAGE STEP, TOTAL CHANNEL CONDUCTANCE, AND LEAK CURRENT OBEY OHMS LAW Ileak For simplicity, we can combine all of the channels and gradients contributing to resting potential into one circuit containing: a resting potential battery slope = Erest gleak and the total conductance of all channels gtotal Vcommand Erest Vcommand out Ileak + - gleak + - Cmem

Erest in (Vcommand - Erest ) x gleak = Ileak A VOLTAGE CHANGE ACROSS THE MEMBRANE ACTS INDEPENDENTLY ON EACH COMPONENT OF THE MEMBRANE CIRCUIT - 61 mV out gK = 2 nS + - RK = 0.5 G +++ out + --- - EK = - 82 mV in Vm = -71 mV INa = - 22 pA IK = 22 pA out gNa = 0.14 nS RNa = 7.1 G - in in 10 mV

out out - 61 mV IK = 42 pA gK = 2 nS RK = 0.5 G + - + - EK = - 82 mV in + + out - - in Vm = -61 mV INa = -20.6 pA IMPOSE COMMAND VOLTAGE ABOVE RESTING POTENTIAL E +Na = + 85 mV gNa = 0.14 nS RNa = 7.1 G

- E +Na in = + 85 mV VOLTAGE STEP FROM RESTING POTENTIAL INDUCES CAPACITANCE TRANSIENT CURRENT AND STEADY-STATE LEAK CURRENT - 61 mV out gK = 2 nS + - RK = 0.5 G + + out + - - - EK = - 82 mV in Vm = -61 mV g = I TOTAL CHANNEL CONDUCTANCE

gtotal(leak) - E = + 85 mV in 10 mV IC x 2.14 nS = NET STEADY STATE CURRENT FLOW Ileak 21.4 pA DISCHARGE = 10 mV x C ITOTAL x VOLTAGE STEP gNa = 0.14 nS RNa = 7.1 G +Na in V INa = -20.6 pA IK = 42 pA out Ileak = (Vcommand - Vrest ) x gtotal(leak) 0

0 time EFFECT OF VOLTAGE STEP ON CURRENTS AND VOLTAGE AT A DIFFERENT SITE WITHIN NEURON SITE OF COMMAND DIFFERENT SITE Vcom - Erest out + Rmem - Cmem in Rmem Cmem Raxial IC ITOTAL ITOTAL IC Ileak = (Vcommand - Erest ) / Rmem 0 0 time 0 0 Ileak = (Vcommand - Erest )/ time

If Raxial is significant, Ileak and voltage divergence from Erest at different site is less and voltage divergence is delayed by R axial x C time constant (Rmem + Raxial ) 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 axons 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 for the 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 Vclamp Rleak + - + - Cmem Erest patch pipet + out

- Vclamp -+ in 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 ground SIMPLIFIED SCHEMATIC OF A TRANSISTOR AMPLIFIER VB POWER SOURCE Vcommand Iinject ground B Vmembrane C A VA IC IC ~ VB - VA If any changes occur in membrane channels causing new currents and drift of Vm, the voltage clamp very rapidly senses this drift and adjusts current injection to maintain Vm = Vcommand

Iinject CURRENT MONITOR Imem patch pipet -+ bath (grounded) Icap CELL 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 Ohms 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