Voltage Gated Ionic Current - Freie...

Voltage Gated Ionic Current

Eryk Wolski (Free University of Berlin)([email protected])

Supervisor: Wilhelm Huisinga

15.05.03

Eryk Wolski Voltage Gated Ionic Current

Contents

• The Membrane Model

– Current through Resistance∗ Models for Voltage Dependent Gating

– Basics of Ionic Battery– Capacitive current

• The Volt Clamp

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The Membrane Model

C - Capacitance; R - resistance of ion channel; Vr - Nernst Potential;Image Source: Christof Koch Sep-21-97: http://www.klab.caltech.edu/ koch/biophysics-book/

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Current through Resistance

• Current caused by Ion i: Iioni= Vi−V

Ri= −gi ∗ (V − Vi)

• sum of the individual ionic currents: Iion =∑

i−gi(V − Vi)

• Vi - reversal potential at which no current are flowing through channel i.

• gi - conductance. gi = gi ∗ f0

gi are the conductance if all channels are open. f0- fraction of open channels.

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Models for Voltage Dependent GatingIon Channels

• Channels are selective for ions of a particular type.

• Individual channels switch among states very rapidly, often more ra-pidly than detectable by commonly used electrophysiological me-thods.

• Switching probabilities of many channels depend upon membranepotential (voltage dependence) or upon the binding of neurotransmit-ters to the membrane (ligand dependence).

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Models for Voltage Dependent GatingA simple molecular Channel

• Switching between an open O and closed C State

• Law of mass action.C ⇀↽k+

k− O

• This transition is called a uni-molecular process because involve on-ly one channel molecule. It is Reversible what is indicated by thearrows.

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• Law of Mass Action The rate of a process is proportional to theproduct of the concentrations of the molecular species involved inthe process

• [O] = fo = No/N

– fo fraction of the open channels.– N total number of channels– No number of open channels– (similarly, fc and Nc refer to the closed state).

• k+ - rate constant. units s−1

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• flux O → C = j− = k−f0

flux C → O = j+ = k+(1− f0)

• Description of the transition from closed to open channels.

df0

dt= j+ − j− = −k−f0 + k+(1− f0) = −(k− + k+)(f0 − k+

(k− + k+))

– f0 fraction of open channels.

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Lets define:

• Fraction of channels open at the equilibrium (dfdt = 0) at the membra-

ne potential V. f∞ = k+

k−+k+

• Factor determining the half time of reaction. τ = 1/(k+ + k−)

The fraction of open channels satisfies the differential equation.

df0

dt=−(f0 − f∞)

τ

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Models for Voltage Dependent GatingThe Voltage dependent channel

• Show image of Voltage dependent potassium channel.

• Voltage dependent Ionic channels are composed of proteins withcharged amino acid side chains

• The potential difference across the membrane can influence the rateat which the transition form the open to closed state occur.

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Models for Voltage Dependent GatingArrhenius Equations

• Ilia has already introduced the Arrhenius Equation !

• The membrane potential V contributes to the energy barrier for thetransitions O ⇀↽ C.

k+ = k+0 e−αV k− = k−0 e−βV

Where k+0 and k−0 are independent of V.

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f∞ =k+

k− + k+τ =

1k+ + k−

Substitute:k+ = k+

0 e−αV k− = k−0 e−βV

into the expression of f∞ and τ .

S0 =1

α− β

Determines the steepness of thedependency of f∞ on V

V0 =ln(k−0 /k+

0 )β − α

Determines the voltage at whichhalf of the channels are open.

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f∞ = 1

1+e−(V−V0)/S0

it is a sigmoidal function

τ = e(αV )

k−0· 1

1+e−(V−V0)/S0

Sigmoidal function combined with exponential function.

0 100 200 300 400 500

0.0

0.2

0.4

0.6

0.8

1.0

V(mV)

f_in

f

0 100 200 300 400 500

0.0

0.5

1.0

1.5

2.0

2.5

V(mV)

T(m

s)

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Models for Voltage Dependent GatingExcitability and Action Potential

A system is excitable when small perturbations return to steady state,but larger (i.e, above a threshold V0) perturbations cause large transientdeviations away from the steady state.

For the voltage dependent ion channel excitability means that the chan-nel are switching from the open to the closed state or vice versa only ifthe voltage transcends V0. How fast this switching occurs depends onS0.

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Basis of the Ionic Battery

• Equilibrium Potential - where the electrical and osmotic forces arebalanced is given by the Nernst Equation .

• It is derived from the change in Gibbs Free Energy4G. At equilibrium4G is zero. Rearranging gives the Nernst Potential.

4V = VNernst =RT

zFln

[ion]out

[ion]in

Nernst Potential is determined by the concentration gradient acrossthe membrane.

R Gas constant; F Faraday constant; T temperature in kalvin ;z valence of the ion

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Capacitive Current

• Given C-capacitance, Vm - membrane Potential, Q-Charge.

Q = Vm · C

• CurrentI =

dQ

dt

• capacitive current

Ic = CdVm(t)

dt

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The Membrane ModelKirchhoffs Law

The sum of all currents flowing into or out of any electrical node mustbe zero.

0 = −Icap + Iion + Iapp

Iapp - any current that might be applied.

CdV

dt= −

∑gi(V − Vi) + Iapp

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Voltage Clamp

Image Source: Christof Koch Sep-21-97: http://www.klab.caltech.edu/ koch/biophysics-book/

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Voltage Clamp

Equations:

• C dVdt = −fog(V − Vrev) + Iapp Differential equitation of membrane

potential.

• We keep the Potential constant by applying Iapp

Iapp = gfo(V − Vrev)

• df0dt = −(f0−f∞)

τ Gating Equation with f∞ = 1

1+e−(V−V0)/S0and τ =

eV (α+β)/2

2√

k+0 k−0 cosh((V−V0)/2S0)

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Voltage Clamp

df0

dt=−(f0 − f∞)

τ

With g = 2,Vo = −25,So = 5;A = (e(V (α+β)/2))/(2√

k+0 k−0 ) = 10 , V = −40,−30,−20,−10, 10 respectively

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Voltage Clamp


With g = 2; Vrev = −65mV and V = −40,−30,−20,−10, 10 and inserting fo we can compute Iapp.

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Voltage Clamp

A different channel with S0 = 1 and V0 = −15 will react in a different wayto the same experimental conditions.


With g = 2; Vrev = −65mV and V = −40,−30,−20,−10, 10 and inserting fo we can compute Iapp.

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Thank you

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